Worksheet 7

For each $m \times n$ matrix $A$ we can create a function \(\begin{aligned} f_A : \mathbb{R}^n & \to \mathbb{R}^m \\ \mathbf{x} & \mapsto A \mathbf{x} \end{aligned}\) where we identify $(x_1,x_2,\ldots,x_n) \in \mathbb{R}^n$ with the vector $(x_1 \ x_2 \ \ldots \ x_n)^T$ and $(x_1,x_2,\ldots,x_m) \in \mathbb{R}^m$ with the vector $(x_1 \ x_2 \ \ldots \ x_m)^T$.

1. Show that $f_A$ is injective if and only if $\mathcal Z(A) = \lbrace 0 \rbrace$.

2. Show that $f_A$ is surjective if and only if $\mathcal R(A) = \mathbb{R}^m$.

3. Recall that $A$ is orthogonal if $A^T = A^{-1}$. Show that $A$ is orthogonal if and only if $f_A$ preserves distance in $\mathbb{R}^n$. Hint: we can compute distance using the dot product of vectors \(\text{dist}(\mathbf{x},\mathbf{y}) = \sqrt{ \mathbf{x}^T \mathbf{y} }\)