6 Linear Transformations Online Quiz Questions

A linear transformation $T:\mathbb{R}^2\to\mathbb{R}^2$ satisfies $T(1,0)=(2,1)$ and $T(0,1)=(-1,3)$. Which matrix represents $T$ in the standard bases?

A linear transformation has a five-dimensional domain and rank $2$. What is its nullity?

Select all statements that are always true for a linear transformation $T:V\to W$.

For $T:\mathbb{R}^3\to\mathbb{R}^2$ defined by $T(x,y,z)=(x+y,y+z)$, what is $\operatorname{im}T$?

True or false: If $A$ represents $U$ and $B$ represents $T$, then the matrix of $T\circ U$ is $BA$, because $U$ is applied first.

A matrix represents a transformation from $\mathbb{R}^4$ to $\mathbb{R}^2$. How many rows does the matrix have?

Select all statements that correctly describe matrix representations and change of basis.

Let $\mathcal{B}'=\{(1,1),(1,-1)\}$ and let $P_{\mathcal{E}\leftarrow\mathcal{B}'}=\begin{bmatrix}1&1\\1&-1\end{bmatrix}$. If $[\mathbf{v}]_{\mathcal{B}'}=(2,3)^T$, what is $[\mathbf{v}]_{\mathcal{E}}$?

For $T:\mathbb{R}^3\to\mathbb{R}^2$ defined by $T(x,y,z)=(x+y,y+z)$, determine the kernel, image, rank, and nullity. Explain how your results satisfy the rank-nullity theorem.

For $T:\mathbb{R}^2\to\mathbb{R}$ defined by $T(x,y)=x+2y$, which subspace is $\ker T$?

Suppose $T$ is linear, $T(\mathbf{u})=(1,-2)$, and $T(\mathbf{v})=(3,1)$. What is $T(2\mathbf{u}-3\mathbf{v})$?

For every linear transformation, the kernel is a subspace of the transformation's domain.

Let $T:\mathbb{R}^2\to\mathbb{R}^2$ satisfy $T(1,0)=(2,1)$ and $T(0,1)=(-1,3)$. What is the matrix of $T$ relative to the standard bases?

Let $U$ have matrix $A=\begin{bmatrix}1&2\\0&1\end{bmatrix}$ and let $T$ have matrix $B=\begin{bmatrix}2&0\\1&3\end{bmatrix}$. What is the entry in row 1, column 2 of the matrix for $T\circ U$?

Let $\mathcal{E}$ be the standard basis and let $\mathcal{B}'=\{(1,1),(1,-1)\}$. If $P_{\mathcal{E}\leftarrow\mathcal{B}'}=\begin{bmatrix}1&1\\1&-1\end{bmatrix}$ and $[\mathbf{v}]_{\mathcal{B}'}=(2,3)^T$, what is $[\mathbf{v}]_{\mathcal{E}}$?

For $T:\mathbb{R}^3\to\mathbb{R}^2$ defined by $T(x,y,z)=(x+y,y+z)$, which statement is correct?

A linear transformation $T:V\to W$ has $\dim V=5$ and $\operatorname{rank}(T)=3$. What is $\operatorname{nullity}(T)$?

True or false: For linear transformations $U:U_0\to V$ and $T:V\to W$, every vector in $\ker U$ is also in $\ker(T\circ U)$.