Linear Algebra is a branch of mathematics that helps you to solve real-world problems. Linear Algebra is challenging for many students, but this course simplifies and demystifies it for any student who has ever struggled with math in the past.
Introduction
This is the most comprehensive linear algebra online course for credit. It covers everything from the basics of linear algebra to more advanced topics like eigenvalues and eigenvectors. The course is taught by Dr. Gilbert Strang, a professor at MIT who is considered to be one of the leading experts on linear algebra.
Linear Equations
Linear equations are mathematical problems that can be expressed in the form of a line. In other words, linear equations are a way of representing a straight line on a graph. Linear equations are used in many different fields, including physics and engineering.
There are many different types of linear equations, but all have the same basic format. A linear equation can be written in the form of:
y = mx + b
Where y is the dependent variable (the variable that changes based on the value of x), m is the slope of the line (the rate at which y changes as x changes), and b is the y-intercept (the point where the line crosses the y-axis).
To solve a linear equation, you need to find the value of x that makes the equation true. This can be done by using algebra or by graphing the equation.
There are many applications for linear equations in real life. For example, linear equations can be used to calculate how long it will take to reach a certain speed if you know the starting speed and acceleration. Linear equations can also be used to predict future values based on past values (known as trend lines).
Vector Spaces
A vector space is a mathematical concept that allows for the addition and multiplication of vectors. Vectors are mathematical objects that have both a magnitude and a direction. A vector space is a set of vectors that contains the zero vector, which is the vector with a magnitude of zero and no direction. Vector spaces can be infinite or finite-dimensional.
Matrices
A matrix is a rectangular array of numbers that is used to represent linear equations. Matrices can be used to represent any kind of mathematical relationship, whether it is a system of linear equations or a transformation from one vector space to another.
In linear algebra, a matrix is usually denoted by a capital letter. For example, A = [aij] is an m×n matrix, where m and n are the number of rows and columns in the matrix, respectively. The entries in the matrix are called elements or entries. The ijth entry in the matrix A is denoted by aij.
The size or dimensions of a matrix are denoted by m×n, where m is the number of rows and n is the number of columns. For example, the matrix A above is said to be an m×n matrix because it has m rows and n columns. If a matrix has only one row or column, then it is called a row vector or column vector, respectively.
The transpose of a matrix A is denoted by AT and is defined as the matrix whose ijth element is equal to the jith element of A: AT = [aji]. In other words, the transpose of A is obtained by interchange rows and columns in A.
Determinants
Linear algebra is the study of mathematical problems that can be best explained in terms of linear equations. Linear algebra is a critical tool for solving mathematical problems, and the skills learned in this course will be useful in many different fields. This course will cover the following topics:
- Vector spaces and subspaces
- Linear transformations
- Eigenvectors and eigenvalues
- Matrix operations
- Determinants
- Inverse matrices
- Systems of linear equations
- Orthogonality
Eigenvalues and Eigenvectors
An eigenvector of a linear transformation is a nonzero vector that, when that transformation is applied to it, does not change direction. In other words Distance Calculus, after being transformed by the matrix associated with the linear transformation, the eigenvector points in the same direction as before. Eigenvectors and their corresponding eigenvalues are extremely important in physics and engineering applications because they can be used to decouple systems of differential equations. This means that the system can be simplified by looking at each equation separately.
To find an eigenvector of a matrix, we need to solve for the values of x that satisfy the equation:
Ax = λx
where A is our matrix, x is our eigenvector, and λ is our eigenvalue. Let’s go through an example to see how this works in practice. Suppose we have the following matrix:
A = 𝟙2 + 2𝟚2
This matrix has two eigenvectors: (1/√2, 1/√2) and (-1/√2, 1/√2). To find these vectors, we need to solve for the values of x that satisfy the equation:
(𝟙2 + 2𝟚2)x = λx
We can do this by setting up a system of equations and solving for x:
𝟙2×1 + 2 𝟚2×2 = λx1 \ 𝟙2×1 + 2𝟚2×2 = λx2
We can solve this system of equations using any method we like. For example, we could use substitution or elimination. In this case, we will use Gaussian elimination. We start by subtracting the first equation from the second equation:
0𝟙2×1 + 0𝟚2×2 = 0x1 – 𝟙2×1 – 2𝟚2×2 = (λ – 𝟙)x2
Applications of Linear Algebra
Linear algebra is the study of mathematical problems that can be best explained in terms of linear equations. Linear algebra is a powerful tool for solving mathematical problems, and it has many applications in the real world.
One application of linear algebra is in the field of computer graphics. Computer graphics use linear equations to create images on a screen. Without linear algebra, computer graphics would not be possible.
Another application of linear algebra is in the field of physics. Physics uses linear equations to describe the motion of objects. Without linear algebra, physics would not be possible.
yet another example where we use Linear Algebra daily without even realizing it is Google Maps! When you type in an address or destination, algorithms are used to calculate the best route by considering various constraints like traffic, distance, and time. All these calculations are based on Linear Algebra!
