Changing Representations of Vectors

In $\mathbb{R}^{2}$, where

find the change of basis matrices from $D$ to ${\cal E}_{2}$ and from ${\cal E}_{2}$ to $D$. Multiply the two.

Which of these matrices could be used to change bases?

• (a)  $\begin{pmatrix}1&2\\ 3&4\end{pmatrix}$     (b)  $\begin{pmatrix}0&-1\\ 1&-1\end{pmatrix}$     (c)  $\begin{pmatrix}2&3&-1\\ 0&1&0\end{pmatrix}$     (d)  $\begin{pmatrix}2&3&-1\\ 0&1&0\\ 4&7&-2\end{pmatrix}$     (e)  $\begin{pmatrix}0&2&0\\ 0&0&6\\ 1&0&0\end{pmatrix}$

Find the change of basis matrix for $B,D\subseteq\mathbb{R}^{2}$.

• (a)  $B={\cal E}_{2}$, $D=\langle\vec{e}_{2},\vec{e}_{1}\rangle$     (b)  $B={\cal E}_{2}$, $D=\langle\begin{pmatrix}1\\ 2\end{pmatrix},\begin{pmatrix}1\\ 4\end{pmatrix}\rangle$     (c)  $B=\langle\begin{pmatrix}1\\ 2\end{pmatrix},\begin{pmatrix}1\\ 4\end{pmatrix}\rangle$, $D={\cal E}_{2}$     (d)  $B=\langle\begin{pmatrix}-1\\ 1\end{pmatrix},\begin{pmatrix}2\\ 2\end{pmatrix}\rangle$, $D=\langle\begin{pmatrix}0\\ 4\end{pmatrix},\begin{pmatrix}1\\ 3\end{pmatrix}\rangle$

Find the change of basis matrix for each $B,D\subseteq\mathcal{P}_{2}$.

• (a)  $B=\langle 1,x,x^{2}\rangle,D=\langle x^{2},1,x\rangle$     (b)  $B=\langle 1,x,x^{2}\rangle,D=\langle 1,1+x,1+x+x^{2}\rangle$     (c)  $B=\langle 2,2x,x^{2}\rangle,D=\langle 1+x^{2},1-x^{2},x+x^{2}\rangle$

For the bases in Exercise 3, find the change of basis matrix in the other direction, from $D$ to $B$.

Decide if each changes bases on $\mathbb{R}^{2}$. To what basis is ${\cal E}_{2}$ changed?

• (a)  $\begin{pmatrix}5&0\\ 0&4\end{pmatrix}$     (b)  $\begin{pmatrix}2&1\\ 3&1\end{pmatrix}$     (c)  $\begin{pmatrix}-1&4\\ 2&-8\end{pmatrix}$     (d)  $\begin{pmatrix}1&-1\\ 1&1\end{pmatrix}$

For each space find the matrix changing a vector representation with respect to $B$ to one with respect to $D$.

• $V=\mathbb{R}^{3}$, $B={\cal E}_{3}$, $D=\langle\begin{pmatrix}1\\ 2\\ 3\end{pmatrix},\begin{pmatrix}1\\ 1\\ 1\end{pmatrix},\begin{pmatrix}0\\ 1\\ -1\end{pmatrix}\rangle$

• $V=\mathbb{R}^{3}$, $B=\langle\begin{pmatrix}1\\ 2\\ 3\end{pmatrix},\begin{pmatrix}1\\ 1\\ 1\end{pmatrix},\begin{pmatrix}0\\ 1\\ -1\end{pmatrix}\rangle$, $D={\cal E}_{3}$

• $V=\mathcal{P}_{2}$, $B=\langle x^{2},x^{2}+x,x^{2}+x+1\rangle$, $D=\langle 2,-x,x^{2}\rangle$

Find bases such that this matrix represents the identity map with respect to those bases.

Consider the vector space of real-valued functions with basis $\langle\sin(x),\cos(x)\rangle$. Show that $\langle 2\sin(x)+\cos(x),3\cos(x)\rangle$ is also a basis for this space. Find the change of basis matrix in each direction.

What is the change of basis matrix with respect to $B,B$?

Prove that a matrix changes bases if and only if it is invertible.

Finish the proof of Lemma 1.5.

Let $H$ be an $n\!\times\!n$ nonsingular matrix. What basis of $\mathbb{R}^{n}$ does $H$ change to the standard basis?

• In $\mathcal{P}_{3}$ with basis $B=\langle 1+x,1-x,x^{2}+x^{3},x^{2}-x^{3}\rangle$ we have this representation.

  Find a basis $D$ giving this different representation for the same polynomial.

• State and prove that we can change any nonzero vector representation to any other.

Hint. The proof of Lemma 1.5 is constructive — it not only says the bases change, it shows how they change.

Let $V,W$ be vector spaces, and let $B,\hat{B}$ be bases for $V$ and $D,\hat{D}$ be bases for $W$. Where $h\colon V\to W$ is linear, find a formula relating ${\rm Rep}_{B,D}(h)$ to ${\rm Rep}_{\hat{B},\hat{D}}(h)$.

Show that the columns of an $n\!\times\!n$ change of basis matrix form a basis for $\mathbb{R}^{n}$. Do all bases appear in that way: can the vectors from any $\mathbb{R}^{n}$ basis make the columns of a change of basis matrix?

Find a matrix having this effect.

That is, find a $M$ that left-multiplies the starting vector to yield the ending vector. Is there a matrix having these two effects?

• (a)  $\begin{pmatrix}1\\ 3\end{pmatrix}\mapsto\begin{pmatrix}1\\ 1\end{pmatrix}\quad\begin{pmatrix}2\\ -1\end{pmatrix}\mapsto\begin{pmatrix}-1\\ -1\end{pmatrix}$     (b)  $\begin{pmatrix}1\\ 3\end{pmatrix}\mapsto\begin{pmatrix}1\\ 1\end{pmatrix}\quad\begin{pmatrix}2\\ 6\end{pmatrix}\mapsto\begin{pmatrix}-1\\ -1\end{pmatrix}$

Give a necessary and sufficient condition for there to be a matrix such that $\vec{v}_{1}\mapsto\vec{w}_{1}$ and $\vec{v}_{2}\mapsto\vec{w}_{2}$.