Basis with Respect to Which the Matrix for Linear Transformation is Diagonal

Problem 315

Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by
$T(ax+b)=(3a+b)x+a+3,$ for any $ax+b\in P_1$.

(a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation $T$.

(b) Find a basis $B’$ of the vector space $P_1$ such that the matrix of $T$ with respect to $B’$ is a diagonal matrix.

(c) Express $f(x)=5x+3$ as a linear combination of basis vectors of $B’$.

(a) Find the matrix of the linear transformation $T$ with respect to the basis $B=\{1, x\}$.

The matrix $A$ for $T$ with respect to the basis $B=\{1, x\}$ is given by
$A=[\, [T(1)]_B, [T(x)]_B\,].$ Since we have
$T(1)=x+3 \text{ and } T(x)=3x+1,$ the coordinate vectors of these vector with respect to $B$ are
\begin{align*}
[T(1)]_B=\begin{bmatrix}
3 \\
1
\end{bmatrix} \text{ and } [T(x)]_B=\begin{bmatrix}
1 \\
3
\end{bmatrix}.
\end{align*}
Therefore, the matrix $A$ for $T$ with respect to $B$ is
$A=\begin{bmatrix} 3 & 1\\ 1& 3 \end{bmatrix}.$

(b) Find a basis $B’$ of the vector space $P_1$ such that the matrix of $T$ with respect to $B’$ is a diagonal matrix.

We first diagonalize the matrix $A$ obtained in (a).
Solving the characteristic polynomial
\begin{align*}
p_A(t)=\det(A-tI)=\begin{bmatrix}
3-t & 1\\
1& 3-t
\end{bmatrix}\\
=t^2-6t+8=(t-2)(t-4),
\end{align*}
the eigenvalues are $2$ and $4$. Since $A$ has two distinct eigenvalues, it is diagonalizable.

We find the eigenvectors corresponding to the eigenvalue $2$.
The eigenvectors are nonzero solutions of $(A-2I)\mathbf{x}=\mathbf{0}$.
From this, we find that $\begin{bmatrix} 1 \\ -1 \end{bmatrix}$ is an eigenvector.
Similarly, we find that $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ is an eigenvector corresponding to the eigenvalue $4$.

Thus, if we set
$P=\begin{bmatrix} 1 & 1\\ -1& 1 \end{bmatrix},$ then we have the diagonalization of $A$:
$\begin{bmatrix} 2 & 0\\ 0& 4 \end{bmatrix}=P^{-1}AP.$ It follows that $\begin{bmatrix} 2 & 0\\ 0& 4 \end{bmatrix}$ is the matrix representation for $T$ with respect to the basis
$B’=\{\mathbf{v}_1, \mathbf{v}_2\},$ where
$[\mathbf{v}_1]_B=\begin{bmatrix} 1 \\ -1 \end{bmatrix} \text{ and } [\mathbf{v}_2]_B=\begin{bmatrix} 1 \\ 1 \end{bmatrix},$ and $P$ is the transition matrix (change of basis) from $B’$ to $B$.

(Note that
$[\mathbf{v}_1]_B=P[\mathbf{v}_1]_{B’}=P\begin{bmatrix} 1 \\ 0 \end{bmatrix}=\begin{bmatrix} 1 \\ -1 \end{bmatrix}.$ Similarly for $[\mathbf{v}_2]_B$.)

Hence, $\mathbf{v}_1=-x+1$ and $\mathbf{v}_2=x+1$ form the basis $B’=\{\mathbf{v}_1, \mathbf{v}_2\}$ with respect to which the matrix for $T$ is the diagonal matrix $\begin{bmatrix} 2 & 0\\ 0& 4 \end{bmatrix}$.

(c) Express $f(x)=5x+3$ as a linear combination of basis vectors of $B’$.

Note that since $P$ is the transition matrix from $B’$ to $B$, we have the following relation between coordinate vectors. For any $\mathbf{v}\in P_1$, we have
$P[\mathbf{v}]_{B’}=[\mathbf{v}]_B.$ Using this, we have
\begin{align*}
[f(x)]_{B’}&=P^{-1}[f(x)]_B\\
&=\frac{1}{2}\begin{bmatrix}
1 & -1\\
1& 1
\end{bmatrix}\begin{bmatrix}
3 \\
5
\end{bmatrix}=\begin{bmatrix}
-1 \\
4
\end{bmatrix}.
\end{align*}
Hence we obtain the linear combination of $f(x)$ with $\mathbf{v}_1, \mathbf{v}_2$:
\begin{align*}
f(x)&=(-1)\mathbf{v}_1+4\mathbf{v}_2\\
\end{align*}

(You may confirm that $5x+3=-(-x+1)+4(x+1)$.)

Let $T$ be the linear transformation from the vector space $\R^2$ to $\R^2$ itself given by \[T\left( \begin{bmatrix} x_1 \\...