Exercises

01

Projection and inclusion

Suppose $p:{\mathbb{R}}^4\to {\mathbb{R}}^2$ and $i:{\mathbb{R}}^2\to {\mathbb{R}}^4$ are two linear mappings given by the formulas:

$$p:\left(\begin{array}{c}{u}_1\\ u_2\\ u_3\\ u_4\end{array}\right)\mapsto \left(\begin{array}{c}{u}_3\\ u_4\end{array}\right),i:\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)\mapsto \left(\begin{array}{c}{v}_1\\ v_2\\ 0\\ 0\end{array}\right).$$

• (a) Check that $p$ and $i$ (given by these formulas) are linear mappings.
• (b) Compute the matrices that correspond to $p$ and $i$.
• (c) Compute the matrix of the composition $i\circ p$ by evaluating the effect of this composite function upon the standard basis vectors ${\mathbf{e}}_1,\dots ,{\mathbf{e}}_4$.
• (d) Compute the same matrix by multiplying the matrix of $i$ and the matrix of $p$.
• (e) Repeat (c) and (d) for $p\circ i$. (This time you only need ${\mathbf{e}}_1,{\mathbf{e}}_2$.)

02

Change of basis using row reduction

Find the change of basis transfer matrices between the following two bases:

$$\mathcal{B}=\left\{{\mathbf{b}}_1=\left(\begin{array}{c}-6\\ -1\end{array}\right),{\mathbf{b}}_2=\left(\begin{array}{c}2\\ 0\end{array}\right)\right\},\mathcal{C}=\left\{{\mathbf{c}}_1=\left(\begin{array}{c}2\\ -1\end{array}\right),{\mathbf{c}}_2=\left(\begin{array}{c}6\\ -2\end{array}\right)\right\}.$$

03

Changing basis without knowing vectors

Assume that $\mathcal{B}=\{{\mathbf{b}}_1,{\mathbf{b}}_2,{\mathbf{b}}_3\}$ and $\mathcal{C}=\{{\mathbf{c}}_1,{\mathbf{c}}_2,{\mathbf{c}}_3\}$ are bases of ${\mathbb{R}}^3$, and that

$$\begin{array}{rl}{\mathbf{b}}_1& =4{\mathbf{c}}_1-{\mathbf{c}}_2\\ {\mathbf{b}}_2& =-{\mathbf{c}}_1+{\mathbf{c}}_2+{\mathbf{c}}_3\\ {\mathbf{b}}_3& ={\mathbf{c}}_2-2{\mathbf{c}}_3.\end{array}$$

Find the change of basis transfer matrix ${}_{\mathcal{C}}{T}_{\mathcal{B}}$, and also compute $[\mathbf{x}{]}_{\mathcal{C}}$ for the vector $\mathbf{x}=3{\mathbf{b}}_1+4{\mathbf{b}}_2+{\mathbf{b}}_3$.

04

Transpose has ‘reversing property’

Show that $(AB{)}^{\mathsf{T}}=B^{\mathsf{T}}{A}^{\mathsf{T}}$.

Hint: You should use the summation notation for matrix products: $(AB{)}_{ij}={\sum }_{k=1}^n{a}_{i\ell }{b}_{\ell j}.$