Learn Linear Algebra

Theorem

Let $T: \mathbb{R}^n \mapsto \mathbb{R}^n$ be a linear transformation. Then $T$ is one-to-one if and only if the equation $T(x)=0$ has only the trivial solution.

Proof

Theorem

Let $T: \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation and let $A$ be the standard matrix for $T$. Then:

a. $T$ maps $\mathbb{R}^n$ onto $\mathbb{R}^m$ if and only if the columns of $A$ span $\mathbb{R}^m$;

b. $T$ is one-to-one if and only if the columns of $A$ are linearly independent.

Proof

Theorem

Let $T: \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Then there exists a unique matrix $A$ such that $$T(x) = Ax \text{ for all } x \in \mathbb{R}^n.$$ In fact, $A$ is the $m \times n$ matrix whose $j$-th column is the vector $T(e_j)$, where $e_j$ is the $j$-th column of the identity
matrix in $\mathbb{R}^n$: $$A = [T(e_1) \ \cdots \ T(e_n)].$$

Proof