Linear Algebra

I assume you are familiar with the vector space \mathbf{R}^n = \{(x_1,\ldots,x_n)\mid x_j\in\mathbf{R}\}, where \mathbf{R} denotes the real numbers, which has the dot product x\cdot y = \sum_j x_j y_j, x,y\in\mathbf{R}^n. We also use the notation (x,y) for x\cdot y. You might also see this written as x'y or x^Ty where the superscript indicates transpose. The dot product defines a norm \|x\| = \sqrt{(x,x)}, or length of a vector.

Exercise. Show \|x\| = 0 if and only if x = 0, |tx| = |t||x|,|(x,y)| |x| |y|, and|x + y| |x| + |y|$, x,y\in\mathbf{R}^n, t\in\mathbf{R}.

Hint. 0 \le \|tx + y\|^2 = t^2\|x\|^2 + 2t(x, y) + \|y\|^2 for t\in\mathbf{R} so (x,y)^2 - \|x\|^2\|y\|^2 \le 0.

This also shows |(x,y)| = \|x\| \|y\| if and only if y = tx for some t\in\mathbf{R} when x and y are not zero.

We want to describe all linear operators T\colon\mathbf{R}^n\to\mathbf{R}^n. A linear operator is a function that preserves the vector space operations of \mathbf{R}^n: T(tx) = t(Tx) and T(x + y) = Tx + Ty, x,y\in\mathbf{R}^n, t\in\mathbf{R}.

Exercise. If n = 1 and t = T1 then Tx = tx for all x\in\mathbf{R}^1.

Let (e_j^n)_{j=1}^n be the standard basis of \mathbf{R}^n, where the j-th component of e_j is 1 and all others are 0. Write e_j if n is understood. In the exercise above we identified e_1 = e_1^1 = (1) with the real number 1. Every element of \mathbf{R}^1 can be written t e_1 for some t\in\mathbf{R}. This shows \mathbf{R}^1 is the “same as” \mathbf{R}, but it is a good idea to keep track of when you make this identification.

Every vector x = (x_1,\ldots,x_n)\in\mathbf{R}^n can be written using the standard basis as x = \sum_j x_j e_j. Every linear operator T\colon\mathbf{R}^n\to\mathbf{R}^n is determined by its values on the standard basis. If Te_j = f_j\in\mathbf{R}^n then Tx = T(\sum_j x_j e_j) = \sum_j x_j f_j, x\in\mathbf{R}^n. Each f_j = \sum_k t_{jk} e_k for some t_{jk}\in\mathbf{R}. We say (t_{jk})_{j,k=1}^n is the _matrix representation of T. This identifies all linear transformations on \mathbf{R}^n with \mathbf{R}^{n^2}.

Matrix multiplication is composition…

n = 2

Se_1 = e_2, Se_2 = 0

p(S) = p(0)I + p(1)S.

General n. p(S) \sum_{j=0}^n p^{(j)}S^j/j!.

We can define f(T) for any f.