In Example CBCV we computed the vector representation of $\vect{y}$ relative to $C$, $\vectrep{C}{\vect{y}}$, as an example of Theorem CB. Compute this same representation directly. In other words, apply Definition VR rather than Theorem CB.

Perform a check on Example MRCM by computing $\matrixrep{Q}{B}{D}$ directly. In other words, apply Definition MR rather than Theorem MRCB.

Find a basis for the vector space $P_3$ composed of eigenvectors of the linear transformation $T$. Then find a matrix representation of $T$ relative to this basis. \begin{equation*} \ltdefn{T}{P_3}{P_3},\quad\lt{T}{a+bx+cx^2+dx^3}= (a+c+d)+(b+c+d)x+(a+b+c)x^2+(a+b+d)x^3 \end{equation*}

Let $S_{22}$ be the vector space of $2\times 2$ symmetric matrices. Find a basis $C$ for $S_{22}$ that yields a diagonal matrix representation of the linear transformation $R$. \begin{align*} \ltdefn{R}{S_{22}}{S_{22}},\quad \lt{R}{\begin{bmatrix}a&b\\b&c\end{bmatrix}}= \begin{bmatrix} -5a + 2b - 3c & -12a + 5b - 6c\\ -12a + 5b - 6c & 6a - 2b + 4c \end{bmatrix} \end{align*}

Let $S_{22}$ be the vector space of $2\times 2$ symmetric matrices. Find a basis for $S_{22}$ composed of eigenvectors of the linear transformation $\ltdefn{Q}{S_{22}}{S_{22}}$. \begin{equation*} \lt{Q}{ \begin{bmatrix} a & b\\ b & c \end{bmatrix} } = \begin{bmatrix} 25a + 18b + 30c & -16a - 11b - 20c\\ -16a - 11b - 20c & -11a - 9b - 12c \end{bmatrix} \end{equation*}

Suppose that $\ltdefn{T}{V}{V}$ is an invertible linear transformation with a nonzero eigenvalue $\lambda$. Prove that $\displaystyle\frac{1}{\lambda}$ is an eigenvalue of $\ltinverse{T}$.

Suppose that $V$ is a vector space and $\ltdefn{T}{V}{V}$ is a linear transformation. Prove that $T$ is injective if and only if $\lambda=0$ is not an eigenvalue of $T$.