# Introduction to linear algebra — pycse (2023)

• KEYWORDS: numpy.transpose, numpy.eye, numpy.diag, numpy.tri, @, numpy.transpose, numpy.allclose, numpy.linalg.det, numpy.linalg.inv, numpy.linalg.matrix_rank, numpy.linalg.cond, numpy.linalg.solve

Just a reminder of where we are! We are now in the Plains of Linear Algebra.

## Multidimensional arrays#

The foundation of linear algebra in Python is in multidimensional arrays.

import numpy as np

We make multidimensional arrays by using lists of lists of numbers. For example, here is a 2D array:

A = np.array([[1, 2], [3, 4]])

We can find out the shape of an array, i.e. the number of rows and columns from the shape attribute. It returns (rows, columns).

A.shape, A.size, len(A)
((2, 2), 4, 2)
for row in A: print(row)
[1 2][3 4]
A[1]
array([3, 4])

### Constructing arrays#

You can always make arrays by typing them in. There are many convenient ways to make special ones though. For example, you can make an array of all ones or zeros with these:

np.zeros(shape=[3, 3])
array([[0., 0., 0.], [0., 0., 0.], [0., 0., 0.]])
np.ones(shape=[3, 3])
array([[1., 1., 1.], [1., 1., 1.], [1., 1., 1.]])

You can make an identity matrix with:

np.eye(N=4)
array([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]])

or a diagonal array:

I = np.eye(N=3, dtype=int)I[1, 1] = 2I[2, 2] = 3I
array([[1, 0, 0], [0, 2, 0], [0, 0, 3]])
np.diag([1, 2, 3])* 1.0
array([[1., 0., 0.], [0., 2., 0.], [0., 0., 3.]])

If you need a lower triangular array:

np.ones((3, 3)) - np.tri(3) + np.eye(3)
array([[1., 1., 1.], [0., 1., 1.], [0., 0., 1.]])
np.triu(np.ones(3)) # How to get an upper triangle array
array([[1., 1., 1.], [0., 1., 1.], [0., 0., 1.]])

### Regular Algebra with arrays#

It takes some getting use to how to use arrays with algebra.

#### Addition and subtraction#

Let’s start with addition and subtraction. A good rule to remember that you can add and subtract arrays with the same shape.

A
array([[1, 2], [3, 4]])
B = np.ones(A.shape)A + B
array([[2., 3.], [4., 5.]])
A - B
array([[0., 1.], [2., 3.]])

This is an error though because the shapes do not match.

C = np.array([[0, 0, 1], [1, 0, 0]])A + C
---------------------------------------------------------------------------ValueError Traceback (most recent call last)Cell In[16], line 4 1 C = np.array([[0, 0, 1], 2 [1, 0, 0]])----> 4 A + CValueError: operands could not be broadcast together with shapes (2,2) (2,3)

Note, however, that this is ok. This feature is called broadcasting. It works when the thing you are adding can be added to each row.

C * np.array([2, 2, 2])
array([[0, 0, 2], [2, 0, 0]])
np.array([[[1], [1]]]).shape
(1, 2, 1)

Exercise Use some algebra to get an array that is ones above the main diagonal, and zeros everywhere else.

np.triu(np.ones(3), k=1)
array([[0., 1., 1.], [0., 0., 1.], [0., 0., 0.]])
np.ones((3, 3)) - np.tri(3)
array([[0., 1., 1.], [0., 0., 1.], [0., 0., 0.]])

#### Multiplication and division#

The default multiplication and division operators work element-wise.

A
array([[1, 2], [3, 4]])
2 * A
array([[2, 4], [6, 8]])
1 / A
array([[1. , 0.5 ], [0.33333333, 0.25 ]])
A * B
array([[1., 2.], [3., 4.]])
B / A
array([[1. , 0.5 ], [0.33333333, 0.25 ]])
A % 2
array([[1, 0], [1, 0]])
A**2
array([[ 1, 4], [ 9, 16]])
np.sqrt(A)
array([[1. , 1.41421356], [1.73205081, 2. ]])

### Matrix algebra#

To do matrix multiplication you use the @ operator (This is new in Python 3.5), or the numpy.dot function. If you are not familiar with the idea of matrix multiplication you should review it at https://en.wikipedia.org/wiki/Matrix_multiplication.

We write matrix multiplication as: $$\mathbf{A} \mathbf{B}$$. We cannot multiply any two arrays; their shapes must follow some rules. We can multiply any two arrays with these shapes:

(m, c) * (c, n) = (m, n)

In other words the number of columns in the first array must equal the number of rows in the second array. This means it is not generally true that $$\mathbf{A} \mathbf{B} = \mathbf{B} \mathbf{A}$$.

A.shape, B.shape
((2, 2), (2, 2))
A
array([[1, 2], [3, 4]])
B
array([[1., 1.], [1., 1.]])
A @ B
array([[3., 3.], [7., 7.]])

This is the older way to do matrix multiplication.

np.dot(A, B)
array([[3., 3.], [7., 7.]])

These rules are true:

1. $$(k \mathbf{A})\mathbf{B} = k(\mathbf{A} \mathbf{B}) = \mathbf{A}(k\mathbf{B})$$

2. $$\mathbf{A}(\mathbf{B}\mathbf{C}) = (\mathbf{A}\mathbf{B})\mathbf{C}$$

3. $$(\mathbf{A} + \mathbf{B})\mathbf{C} = \mathbf{A}\mathbf{B} + \mathbf{A}\mathbf{C}$$

4. $$\mathbf{C}(\mathbf{A} + \mathbf{B}) = \mathbf{C}\mathbf{A} + \mathbf{C}\mathbf{B}$$

Exercise construct examples of each of these rules.

We can also multiply a matrix and vector. This is like the shapes of (m, r) * (r, 1) = (m, 1)

k = 2(k * A) @ B == k * (A @ B)
array([[ True, True], [ True, True]])
# Checking rule #4C = np.random.random((2, 2))Cnp.allclose((C @ (A + B)), (C @ A + C @ B))
True
x = np.array([1, 2])A @ x
array([ 5, 11])

There is a small subtle point, the x-array is 1-D:

x.shape
(2,)

Its shape is not (2, 1)! Numpy does the right thing here and figures out what you want. Not all languages allow this, however, and you have to be careful that everything has the right shape with them.

## Linear algebra functions of arrays#

### The transpose#

In the transpose operation you swap the rows and columns of an array. The transpose of A is denoted $$\mathbf{A}^T$$.

A
array([[1, 2], [3, 4]])
A.T
array([[1, 3], [2, 4]])

There is also a function for transposing.

np.transpose(A)
array([[1, 3], [2, 4]])

A matrix is called symmetric if it is equal to its transpose: $$\mathbf{A} == \mathbf{A}^T$$.

Q = np.array([[1, 2], [2, 4]])np.allclose(Q, Q.T)
True

A matrix is called skew symmetric if $$\mathbf{A}^T = -\mathbf{A}$$.

Q = np.array([[0, 1], [-1, 0]])np.allclose(Q.T, -Q)
True

A matrix is called orthogonal if this equation is true: $$\mathbf{A} \mathbf{A}^T = \mathbf{I}$$. Here is an example of an orthogonal matrix:

theta = 120Q = np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])with np.printoptions(suppress=True): print(Q @ Q.T)
[[1. 0.] [0. 1.]]

Here are the four rules for matrix multiplication and transposition

1. $$(\mathbf{A}^T)^T = \mathbf{A}$$

2. $$(\mathbf{A}+\mathbf{B})^T = \mathbf{A}^T+\mathbf{B}^T$$

3. $$(\mathit{c}\mathbf{A})^T = \mathit{c}\mathbf{A}^T$$

4. $$(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$$

Exercise Come up with an example for each rule.

# rule #4A = np.random.random((30, 30))B = np.random.random((30, 30))np.max(np.abs((A @ B).T - B.T @ A.T))
0.0

### The determinant#

The determinant of a matrix is noted: det(A) or |A|. Many matrices are used to linearly transform vectors, and the determinant is related to the scaling magnitude.

np.linalg.det(A)
6.043051860665325

### The inverse#

A matrix is invertible if and only if the determinant of the matrix is non-zero.

The inverse is defined by: $$\mathbf{A} \mathbf{A}^{-1} = \mathbf{I}$$.

We compute the inverse as:

A = np.random.random((3, 3))np.linalg.inv(A)
array([[ 1.88528843, -2.5603516 , 3.75249338], [-3.15258757, 0.11414811, 2.9170395 ], [ 1.60379253, 1.5751048 , -2.37883069]])

And here verify the definition.

with np.printoptions(suppress=True): print(A @ np.linalg.inv(A))
[[ 1. 0. -0.] [ 0. 1. -0.] [-0. 0. 1.]]

Another way to define an orthogonal matrix is $$\mathbf{A}^T = \mathbf{A}^{-1}$$.

A A.T = I

A.inv A A.T = A.inv I = A.inv

I A.T = A.inv

A.T = A.inv

theta = 12Q = np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])np.allclose(Q.T, np.linalg.inv(Q))
True

### Rank#

The rank of a matrix is equal to the number of linearly independent rows in it. Rows are linearly independent if and only if they cannot be made by constants times another row or linear combinations of other rows.

np.linalg.matrix_rank(A)
3

Here is an example of a rank-deficient array. The last row is a linear combination of the first two rows.

A1 = [[1, 2, 3], [0, 2, 3], [2, 6, 9]]np.linalg.matrix_rank(A1)
2

Here is an example of a rank-deficient array. It is deficient because the last row is just 0 times any other row.

A1 = [[1, 2, 3], [0, 2, 3], [0, 0, 0]]np.linalg.matrix_rank(A1)
2

Note the determinant of this array is nearly zero as a result.

np.linalg.det(A1)
0.0

Also note the inverse may be undefined or have some enormous numbers in it. This is not a reliable inverse. It is never a good idea to have giant numbers and small numbers in the same calculations!

np.linalg.inv(A1)
---------------------------------------------------------------------------LinAlgError Traceback (most recent call last)Cell In[53], line 1----> 1 np.linalg.inv(A1)File <__array_function__ internals>:200, in inv(*args, **kwargs)File /opt/hostedtoolcache/Python/3.8.16/x64/lib/python3.8/site-packages/numpy/linalg/linalg.py:538, in inv(a) 536 signature = 'D->D' if isComplexType(t) else 'd->d' 537 extobj = get_linalg_error_extobj(_raise_linalgerror_singular)--> 538 ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj) 539 return wrap(ainv.astype(result_t, copy=False))File /opt/hostedtoolcache/Python/3.8.16/x64/lib/python3.8/site-packages/numpy/linalg/linalg.py:89, in _raise_linalgerror_singular(err, flag) 88 def _raise_linalgerror_singular(err, flag):---> 89 raise LinAlgError("Singular matrix")LinAlgError: Singular matrix

The condition number is a measure of the norm of an array times the inverse of the array. If it is very large, the array is said to be ill-conditioned.

np.linalg.cond(A1)
inf

What all of these mean is that we only have two independent rows in the array.

## Solving linear algebraic equations#

One of the key reasons to develop the tools above is for solving linear equations. Let’s consider an example.

Given these equations, find [x1, x2, x3]

$$x_1 - x_2 + x_3 = 0$$

$$10 x_2 + 25 x_3 = 90$$

$$20 x_1 + 10 x_2 = 80$$

reference: Kreysig, Advanced Engineering Mathematics, 9th ed. Sec. 7.3

First, we express this in the form $$\mathbf{A} \mathbf{x} = \mathbf{b}$$.

A = np.array([[1, -1, 1], [0, 10, 25], [20, 10, 0]])b = np.array([0, 90, 80])

Now, if we left multiply by $$\mathbf{A}^{-1}$$ then we get:

$$\mathbf{A}^{-1} \mathbf{A} \mathbf{x} = \mathbf{A}^{-1} \mathbf{b}$$ which simplifies to:

$$\mathbf{x} = \mathbf{A}^{-1} \mathbf{b}$$

How do we know if there should be a solution? First we make the augmented matrix $$\mathbf{A} | \mathbf{b}$$. Note for this we need \mathbf{b} as a column vector. Here is one way to make that happen. We make it a row in a 2D array, and transpose that to make it a column.

Awiggle = np.hstack([A, np.array([b]).T])Awiggle
array([[ 1, -1, 1, 0], [ 0, 10, 25, 90], [20, 10, 0, 80]])

If the rank of $$\mathbf{A}$$ and the rank of $$\mathbf{\tilde{A}}$$ are the same, then we will have one unique solution. if the rank is less than the number of unknowns, there maybe an infinite number of solutions.

np.linalg.matrix_rank(A), np.linalg.matrix_rank(Awiggle)
(3, 3)

If $$\mathbf{b}$$ is not all zeros, we can also use the fact that a non-zero determinant leads to a unique solution.

np.linalg.det(A)
-950.0000000000001

It should also be evident that since we use an inverse matrix, it must exist (which is certain since the determinant is non-zero). Now we can evaluate our solution.

x = np.linalg.inv(A) @ bx
array([2., 4., 2.])

Now you might see why we vastly prefer linear algebra to nonlinear algebra; there is no guessing or iteration, we just solve the equations!

Let us confirm our solution:

A @ x == b
array([False, True, False])

This fails because of float tolerances:

[float(z) for z in A @ x - b] # subtle point that sometimes small numbers print as zero
[4.440892098500626e-16, 0.0, 1.4210854715202004e-14]

We should instead see if they are all close. You could roll your own comparison, but we instead leverage numpy.allclose for this comparison.

np.allclose(A @ x, b)
True

The formula we used above to solve for $$\mathbf{x}$$ is not commonly used. It turns out computing the inverse of a matrix is moderately expensive. For small systems it is negligible, but the time to compute the inverse grows as $$N^3$$, and there are more efficient ways to solve these when the number of equations grows large.

import numpy as npimport timet = []I = np.array(range(2, 5001, 50))for i in I: m = np.eye(i) t0 = time.time() np.linalg.inv(m) t += [time.time() - t0]import matplotlib.pyplot as pltplt.plot(I, t)plt.xlabel('N')plt.ylabel('Time to invert (s)');

As usual, there is a function we can use to solve this.

np.linalg.solve(A, b)
array([2., 4., 2.])
t = []I = np.array(range(2, 5001, 500))for i in I: A = np.eye(i) b = np.arange(i) t0 = time.time() np.linalg.solve(A, b) t += [time.time() - t0]plt.plot(I, t)plt.xlabel('N')plt.ylabel('Time to solve Ax=b (s)');

You can see by inspection that solve must not be using an inverse to solve these equations; if it did, it would take much longer to solve them. It is remarkable that we can solve ~5000 simultaneous equations here in about 1 second!

This may seem like a lot of equations, but it isn’t really. Problems of this size routinely come up in solving linear boundary value problems where you discretize the problem into a large number of linear equations that are solved.

## Summary#

Today we introduced many functions used in linear algebra. One of the main applications of linear algebra is solving linear equations. These arise in many engineering applications like mass balances, reaction network analysis, etc. Because we can solve them directly (not iteratively with a guess like with non-linear algebra) it is highly desirable to formulate problems as linear ones where possible.

There are many more specialized routines at https://docs.scipy.org/doc/numpy-1.15.1/reference/routines.linalg.html.

## FAQs

### Introduction to linear algebra — pycse? ›

Linear Algebra from a textbook with traditional lectures can be challenging. Many students in traditional lecture courses do rate Linear Algebra as a more difficult course than Calculus I and Calculus II.

Is Introduction to linear algebra hard? ›

Linear Algebra from a textbook with traditional lectures can be challenging. Many students in traditional lecture courses do rate Linear Algebra as a more difficult course than Calculus I and Calculus II.

Is calculus or linear algebra harder? ›

Calculus is the hardest mathematics subject and only a small percentage of students reach Calculus in high school or anywhere else. Linear algebra is a part of abstract algebra in vector space. However, it is more concrete with matrices, hence less abstract and easier to understand.

Can you do linear algebra in Python? ›

To work with linear algebra in Python, you can count on SciPy, which is an open-source Python library used for scientific computing, including several modules for common tasks in science and engineering. Of course, SciPy includes modules for linear algebra, but that's not all.

What is the introduction of linear algebra? ›

Linear algebra is the study of linear combinations. It is the study of vector spaces, lines and planes, and some mappings that are required to perform the linear transformations. It includes vectors, matrices and linear functions. It is the study of linear sets of equations and its transformation properties.

Which is harder Calc 3 or linear algebra? ›

I found Linear Algebra to be the most difficult because it's more abstract then any other math course I had to take for my Engineering Physics curriculum and you really need to understand everything about it and it's abstractness in order to succeed in that course.

What is harder than calculus? ›

In general, statistics is more vast and covers more topics than calculus. Hence, it is also perceived to be more challenging. Basic or entry-level statistics is much easier as compared to basic level calculus. Advance level statistics is much much harder than advanced level calculus.

What is the hardest math class in school? ›

What is the Hardest Math Class in High School? In most cases, you'll find that AP Calculus BC or IB Math HL is the most difficult math course your school offers. Note that AP Calculus BC covers the material in AP Calculus AB but also continues the curriculum, addressing more challenging and advanced concepts.

What level of math is linear algebra? ›

Due to its broad range of applications, linear algebra is one of the most widely taught subjects in college-level mathematics (and increasingly in high school).

What is the hardest math class in college? ›

Advanced Calculus is the hardest math subject, according to college professors. One of the main reasons students struggle to understand the concepts in Advanced Calculus is because they do not have a good mathematical foundation. Calculus builds on the algebraic concepts learned in previous classes.

### What grade is linear algebra taught? ›

Students who take Algebra 1 in 7th grade can complete Calculus in the 11th grade and take an even more advanced math class, such as college-level Linear Algebra, in grade 12. On the other hand, students who want to jump off the Calculus track have other course options, such as Trigonometry or Statistics.

What majors need linear algebra? ›

Linear Algebra is required for math, physics, engineering, statistics, and economics majors.

Do computer scientists use linear algebra? ›

Linear algebra provides concepts that are crucial to many areas of computer science, including graphics, image processing, cryptography, machine learning, computer vision, optimization, graph algorithms, quantum computation, computational biology, information retrieval and web search.

What is the hardest math? ›

The Continuum Hypothesis is a mathematical problem involving the concept of infinity and the size of infinite sets. It was first proposed by Georg Cantor in 1878 and has remained one of the unsolvable and hardest math problems ever since.

Is linear algebra basic math? ›

Introduction to Linear Algebra Basics

Linear algebra is also central to almost all areas of mathematics like geometry and functional analysis. Its concepts are a crucial prerequisite for understanding the theory behind machine learning, especially if you are working with deep learning algorithms.

Should I learn linear algebra or calculus first? ›

So, for those students wishing to get ahead and get Linear Algebra in their completed column in their academic plan, you do need to complete Calculus II first, which means also completing Calculus I first, even though Linear Algebra has nothing to do with either course.

What is the hardest Calc class in college? ›

Those who have difficulties memorizing and applying new, unrelated mathematical techniques may say that Calculus 2 is the most challenging calculus class. On the other hand, students who struggle with making three-dimensional calculations may argue that Calculus 3 is the most difficult.

What math is higher than calculus? ›

After completing Calculus I and II, you may continue to Calculus III, Linear Algebra, and Differential Equations. These three may be taken in any order that fits your schedule, but the listed order is most common.

Which is harder algebra or trigonometry? ›

These are some of the reasons which make Trigonometry so confusing for students, but in general, it is easier than statistics but harder than algebra and geometry.

Do most students fail calculus? ›

Those concerns were heightened given significant declines in recent standardized math test scores for K-12 California students, most of whom spent 2020-21 in distance learning. CSU Bakersfield reports that 35.5% of students on average have been failing or withdrawing from Calculus 1 since 2018.

### What is the most complex branch of math? ›

Calculus

One of the most complex and highly advanced branches of mathematics, which in fact itself has levels to it, be it Pre-calculus, advanced calculus, Accelerated Multivariable Calculus, differential calculus, integral calculus, etc.

What is the highest level of math in college? ›

A doctoral degree is the highest level of education available in mathematics, often taking 4-7 years to complete. Like a master's degree, these programs offer specializations in many areas, including computer algebra, mathematical theory analysis, and differential geometry.

What is the most failed high school math class? ›

Algebra I is the single most failed course in American high schools. Thirty-three percent of students in California, for example, took Algebra I at least twice during their high school careers. And students of color or those experiencing poverty are overrepresented in this group.

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Math 55 is a two-semester long first-year undergraduate mathematics course at Harvard University, founded by Lynn Loomis and Shlomo Sternberg. The official titles of the course are Honors Abstract Algebra (Math 55a) and Honors Real and Complex Analysis (Math 55b).

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Linear algebra plays an important role to determine unknown quantities. The real-life applications of linear algebra are: For calculation of speed, distance, or time. Used for projecting a three-dimensional view into a two-dimensional plane, handled by linear maps.

Do engineers use linear algebra? ›

Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models.

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What is college-level math? Entry-level math in college is considered the stepping stone to more advanced math. Algebra 1, trigonometry, geometry, and calculus 1 are the basic math classes. Once you have successfully navigated through these courses, you can trail blazed through more advanced courses.

What is the hardest math at Harvard? ›

"Math 55" has gained a reputation as the toughest undergraduate math class at Harvard—and by that assessment, maybe in the world. The course is one many students dread, while some sign up out of pure curiosity, to see what all the fuss is about.

What math class do most college freshmen take? ›

The first math course a student takes depends on his or her background. In most cases, it will be MATH 105 (Calculus I), 106 (Calculus II), 205 (Linear Algebra), or 206 (Multivariable Calculus).

Is linear algebra upper level math? ›

Calculus and linear algebra constitute the foundation for most of modern, upper level mathematics.

### Is linear algebra taught after calculus? ›

Two main courses after calculus are linear algebra and differential equations.

Is linear algebra needed for coding? ›

While coders from other disciplines such as web development and front end development don't need to be linear algebra whizzes, understanding the concepts will help you find and use the right tools for advanced problem solving.

Is there any prerequisites to learn linear algebra? ›

Recommended Prerequisites

To succeed in this course you will need to be comfortable with vectors, matrices, and three-dimensional coordinate systems. This material is presented in the first few lectures of 18.02 Multivariable Calculus, and again here.

What are the 4 types of algebra? ›

FAQs on Algebra

There are five different branches or types of algebra. They are elementary algebra, abstract algebra, advanced algebra, commutative algebra, and linear algebra. All these branches have different formulas, different applications, and different uses in finding out the values of variables.

What math is used most in computer science? ›

Binary mathematics is the heart of the computer and an essential math field for computer programming. For all mathematical concepts, the binary number system uses only two digits, 0 and 1. It simplifies the coding process and is essential for low-level instructions used in hardware programming.

Is quantum computing just linear algebra? ›

Linear algebra is the language of quantum computing. Although you don't need to know it to implement or write quantum programs, it is widely used to describe qubit states, quantum operations, and to predict what a quantum computer does in response to a sequence of instructions.

What math class is best for computer science? ›

Discrete mathematics, linear algebra, number theory, and graph theory are the math courses most relevant to the computer science profession. Different corners of the profession, from machine learning to software engineering, use these types of mathematics.

Has 3X 1 been solved? ›

In 1995, Franco and Pom-erance proved that the Crandall conjecture about the aX + 1 problem is correct for almost all positive odd numbers a > 3, under the definition of asymptotic density. However, both of the 3X + 1 problem and Crandall conjecture have not been solved yet.

What's the answer to x3 y3 z3 K? ›

In mathematics, entirely by coincidence, there exists a polynomial equation for which the answer, 42, had similarly eluded mathematicians for decades. The equation x3+y3+z3=k is known as the sum of cubes problem.

How do you get good at linear algebra? ›

An Intuitive Guide to Linear Algebra
1. Name the course Linear Algebra but focus on things called matrices and vectors.
2. Teach concepts like Row/Column order with mnemonics instead of explaining the reasoning.
3. Favor abstract examples (2d vectors! 3d vectors!) and avoid real-world topics until the final week.

### How long does it take to learn linear algebra? ›

The best available course to learn linear algebra is a collection of 35 lectures by Dr. Gilbert Strang on MIT OCW. This course can take at most one month to complete for a complete beginner. But since most of the folks doing machine learning are aware of introductory linear algebra and matrices.

Who is father of linear algebra? ›

The introduction of linear algebra in the West dates back to the year 1637, when René Descartes develop the concept of coordinates under a geometric approach, known today as Cartesian geometry.

Is linear algebra hard in college? ›

Linear Algebra from a textbook with traditional lectures can be challenging. Many students in traditional lecture courses do rate Linear Algebra as a more difficult course than Calculus I and Calculus II.

Is linear algebra necessary for AI? ›

Linear algebra is a field of mathematics that could be called the mathematics of data. It is undeniably a pillar of the field of machine learning, and many recommend it as a prerequisite subject to study prior to getting started in machine learning.

What is the hardest part of linear algebra? ›

Some of the most challenging elements in linear algebra include: defining mathematical structures using a set of axions, wrapping your head around eigenvectors, and grasping the concepts of abstract vector space and linear independence.

Is it easy to learn Linear Algebra? ›

Linear algebra is probably the easiest and the most useful branch of modern mathematics. Indeed, topics such as matrices and linear equations are often taught in middle or high school.

What level of difficulty is linear algebra? ›

Linear algebra is probably the easiest and the most useful branch of modern mathematics. Indeed, topics such as matrices and linear equations are often taught in middle or high school.

What grade do I learn linear algebra? ›

Linear algebra is standard topic in the college mathematics curricula. It is usually taken by students in their sophomore year. Linear Algebra is required for math, physics, engineering, statistics, and economics majors.

Is linear algebra above calculus? ›

After completing Calculus I and II, you may continue to Calculus III, Linear Algebra, and Differential Equations. These three may be taken in any order that fits your schedule, but the listed order is most common.

How many people fail linear algebra? ›

Linear algebra 1 - about 45% fail rate.

### What is the easiest math level? ›

Which math classes are the easiest? According to a large group of high-schoolers, the easiest math class is Algebra 1. That is the reason why most of the students in their freshman year end up taking Algebra 1. Following Algebra 1, Geometry is the second easiest math course in high school.

Is linear algebra higher than calculus 2? ›

You can't study physics in more than a toy sort of way without calculus. I would rate Linear Algebra as more difficult than Calculus II; Calculus II is “more of the same” (as Calculus I) with more on integration/analytic geometry, etc.

Is calculus needed for linear algebra? ›

The pathways to advanced mathematics courses all begin with linear algebra and multivariable calculus, and the standard prerequisite for most linear algebra and multivariable calculus courses includes two semesters of calculus.

How long will it take to learn linear algebra? ›

Linear Algebra is one of the most important concepts required in machine learning and deep learning. The best available course to learn linear algebra is a collection of 35 lectures by Dr. Gilbert Strang on MIT OCW. This course can take at most one month to complete for a complete beginner.

Is linear algebra in college algebra? ›

Study concepts in college algebra, including the solving of linear, quadratic and other algebraic equations and identities; developing mathematical models; and the graphing of linear, exponential, rational, logarithmic and polynomial functions.

What should I learn first before linear algebra? ›

You might need very very basic intuition for geometry, basic algebra (at least simultaneous equations) and addition, subtraction, multiplication, division... maybe some knowledge of vectors and matrices could help at the beginning but a good linear algebra course should teach you what that is anyway.

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