Questions

01

Finding a matrix using linearity

What is the 2x2 matrix that sends $2{\mathbf{e}}_1-2{\mathbf{e}}_2$ to $\left(\begin{array}{c}-4\\ 2\end{array}\right)$ and $-3{\mathbf{e}}_2$ to $\left(\begin{array}{c}3\\ -6\end{array}\right)$?

(Hint: first determine where ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ are sent using linearity. Then: the matrix is given by using these images as column vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$.)

02

Linearity and quadratic forms

Consider the function $f:{\mathbb{R}}^2\to \mathbb{R}$ given by $f\left(\begin{array}{c}x\\ y\end{array}\right)=xy$. Is this function linear? If not, is it linear in each variable separately?

03

Dual vectors

Suppose $\mathbf{v}\in {\mathbb{R}}^n$ is some vector. Define a function $f_{\mathbf{v}}:{\mathbb{R}}^n\to \mathbb{R}$ by the formula $f_{\mathbf{v}}(\mathbf{x})=\mathbf{v}\cdot \mathbf{x}$. Is this function linear? If so, find a $1\times n$ matrix that represents the action of $f_{\mathbf{v}}$. This matrix is often called the dual vector of $\mathbf{v}$.