Basics of Linear Algebra: Things you are supposed to know before Modern II

There are a few basic things you are supposed to know about Linear Algebra by spring of junior year. Most of this is covered in Math Physics, but that class has a lot of territory to cover, much of it seemingly unrelated to the rest, and as such many people have only very basic knowledge coming out of that class, and much of that is forgotten all too easily. This is a simple guide which illustrates many of the concepts at a basic level to clear up any confusion that may occur going into Modern Physics. The official Wikipedia pages dealling with the subject are: Vector Space Matrices Transpose Matrix Multiplication

Matrices and Vectors

There are three kinds of object you need to deal with. The first two are scalars and vectors, and the third is matrices. As you already know, scalars are just numbers, vectors are lists of numbers (like matrices but one dimension is of length one), and matrices are grids of numbers. These are all "tensors". There are more, higher ranked tensors, like 3-D grids of numbers, but those aren't so useful just now. Here are three generic objects: x = x₁ $\vec{x}=\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix}$ $\hat{A}=\begin{bmatrix} a_{1 1}&a_{1 2}&a_{1 3} \\ a_{2 1}&a_{2 2}&a_{2 3} \\ a_{3 1}&a_{3 2}&a_{3 3} \end{bmatrix}$ Note that matrices are usually denoted by capital letters, and that operators (which are matrices) have hats. Also, rows are the first index, while columns are the second.

Transpose

The transpose of a matrix is simply a matrix with rows transmuted into columns, or vice versa. Another way to put it is that the matrix is reflected about its diagonal (the sequence of entries from top-left to bottom-right). $\hat{A}^{\top}=\begin{bmatrix} a_{1 1}&a_{2 1}&a_{3 1} \\ a_{1 2}&a_{2 2}&a_{3 2} \\ a_{1 3}&a_{2 3}&a_{3 3} \end{bmatrix}$