Define the kernel and the image of a linear map $\alpha$ from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$.

Let $\left\{\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{m}\right\}$ be a basis of $\mathbb{R}^{m}$ and $\left\{\mathbf{f}_{1}, \mathbf{f}_{2}, \ldots, \mathbf{f}_{n}\right\}$ a basis of $\mathbb{R}^{n}$. Explain how to represent $\alpha$ by a matrix $A$ relative to the given bases.

A second set of bases $\left\{\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \ldots, \mathbf{e}_{m}^{\prime}\right\}$ and $\left\{\mathbf{f}_{1}^{\prime}, \mathbf{f}_{2}^{\prime}, \ldots, \mathbf{f}_{n}^{\prime}\right\}$ is now used to represent $\alpha$ by a matrix $A^{\prime}$. Relate the elements of $A^{\prime}$ to the elements of $A$.

Let $\beta$ be a linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{3}$ defined by

$$\beta\left(\begin{array}{l} 1 \\ 1 \end{array}\right)=\left(\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right), \quad \beta\left(\begin{array}{c} 1 \\ -1 \end{array}\right)=\left(\begin{array}{l} 6 \\ 4 \\ 2 \end{array}\right)$$

Either find one or more $\mathbf{x}$ in $\mathbb{R}^{2}$ such that

$$\beta \mathbf{x}=\left(\begin{array}{c} 1 \\ -2 \\ 1 \end{array}\right)$$

or explain why one cannot be found.

Let $\gamma$ be a linear map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ defined by

$$\gamma\left(\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right)=\left(\begin{array}{l} 1 \\ 3 \end{array}\right), \quad \gamma\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right)=\left(\begin{array}{c} -2 \\ 1 \end{array}\right), \quad \gamma\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right)=\left(\begin{array}{l} 0 \\ 1 \end{array}\right)$$

Find the kernel of $\gamma$.