An Introduction to Linear Algebra

MTH 309 Introduction to Linear Algebra

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Each chapter concludes with both proof-writing and computational exercises.

Content: 1. What is linear algebra 2. Introduction to complex numbers 3. The fundamental theorem of algebra and factoring polynomials 4.

MA Introduction to Linear Algebra

Vector spaces 5. Span and bases 6.

Linear maps 7. Eigenvalues and eigenvectors 8. A Matrix can have multiple numbers of rows and columns. Note that a Vector is also a Matrix, but with only one row or one column. The Matrix in the example in the yellow graphic is also a 2- by 3-dimensional Matrix rows x columns.

Below you can see another example of a Matrix along with its notation:. You can think of a Tensor as an array of numbers, arranged on a regular grid, with a variable number of axes.

A Tensor has three indices, where the first one points to the row, the second to the column and the third one to the axis. For example, T points to the second row, the third column, and the second axis. This refers to the value 0 in the right Tensor in the graphic below:. Tensor is the most general term for all of these concepts above because a Tensor is a multidimensional array and it can be a Vector and a Matrix, depending on the number of indices it has. For example, a first-order Tensor would be a Vector 1 index. A second-order Tensor is a Matrix 2 indices and third-order Tensors 3 indices and higher are called Higher-Order Tensors 3 or more indices.

If you multiply, divide, subtract, or add a Scalar to a Matrix, you do so with every element of the Matrix. The image below illustrates this perfectly for multiplication:.

Tutorial Overview

Multiplying a Matrix by a Vector can be thought of as multiplying each row of the Matrix by the column of the Vector. The output will be a Vector that has the same number of rows as the Matrix. The image below shows how this works:. To better understand the concept, we will go through the calculation of the second image.

To get the first value of the resulting Vector 16 , we take the numbers of the Vector we want to multiply with the Matrix 1 and 5 , and multiply them with the numbers of the first row of the Matrix 1 and 3. This looks like this:. We do the same for the values within the second row of the Matrix:. And again for the third row of the Matrix:.

An Intuitive Guide to Linear Algebra

And here is a kind of cheat sheet:. Matrix-Matrix Addition and Subtraction is fairly easy and straightforward. The requirement is that the matrices have the same dimensions and the result is a Matrix that has also the same dimensions.

You just add or subtract each value of the first Matrix with its corresponding value in the second Matrix. See below:. The result will be a Matrix with the same number of rows as the first Matrix and the same number of columns as the second Matrix. It works as follows:.

You simply split the second Matrix into column-Vectors and multiply the first Matrix separately by each of these Vectors. Then you put the results in a new Matrix without adding them up! The image below explains this step by step:. And here is again some kind of cheat sheet:. Matrix Multiplication has several properties that allow us to bundle a lot of computation into one Matrix multiplication. We will discuss them one by one below. We will start by explaining these concepts with Scalars and then with Matrices because this will give you a better understanding of the process.

Scalar Multiplication is commutative but Matrix Multiplication is not. Scalar and Matrix Multiplication are both associative. Scalar and Matrix Multiplication are also both distributive. The Identity Matrix is a special kind of Matrix but first, we need to define what an Identity is. The number 1 is an Identity because everything you multiply with 1 is equal to itself. Therefore every Matrix that is multiplied by an Identity Matrix is equal to itself.

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You can spot an Identity Matrix by the fact that it has ones along its diagonals and that every other value is zero. We previously discussed that Matrix multiplication is not commutative but there is one exception, namely if we multiply a Matrix by an Identity Matrix. The Matrix inverse and the Matrix transpose are two special kinds of Matrix properties. Again, we will start by discussing how these properties relate to real numbers and then how they relate to Matrices. First of all, what is an inverse? A number that is multiplied by its inverse is equal to 1.

Note that every number except 0 has an inverse. If you multiply a Matrix by its inverse, the result is its Identity Matrix. The example below shows what the inverse of Scalars looks like:.

But not every Matrix has an inverse.

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