Review Problems for Exam 2

Let \(V=\text {span}\left (\begin{bmatrix}2\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix}, \begin{bmatrix}-4\\0\\-2\end{bmatrix}\right )\). Recall that the span of a set of vectors in \(\RR ^n\) is a subspace of \(\RR ^n\). From the choices below select ALL sets that can serve as bases for \(V\).
\(\left \{ \begin{bmatrix}2\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix}, \begin{bmatrix}-4\\0\\-2\end{bmatrix}\right \}\) \(\left \{ \begin{bmatrix}2\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix}\right \}\) \(\left \{ \begin{bmatrix}0\\1\\1\end{bmatrix}, \begin{bmatrix}-4\\0\\-2\end{bmatrix}\right \}\) \(\left \{ \begin{bmatrix}2\\0\\1\end{bmatrix}, \begin{bmatrix}-4\\0\\-2\end{bmatrix}\right \}\) \(\left \{ \begin{bmatrix}2\\1\\2\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix}\right \}\)
Let \(\mathcal {B}=\left \{ \begin{bmatrix}1\\1\\0\end{bmatrix}, \begin{bmatrix}1\\2\\0\end{bmatrix}, \begin{bmatrix}0\\1\\2\end{bmatrix}\right \}\) be an ordered basis for \(\RR ^3\). (Mentally verify that \(\mathcal {B}\) is a basis of \(\RR ^3\).)

Let \(\vec {v}=\begin{bmatrix}-1\\-3\\2\end{bmatrix}\) be a vector of \(\RR ^3\) (written with respect to the standard basis of \(\RR ^3\)). Find the coordinate vector for \(\vec {v}\) with respect to \(\mathcal {B}\).

\[\begin{bmatrix}\answer {2}\\\answer {-3}\\\answer {1}\end{bmatrix}\]
Consider matrix \(A\) and \(\text {rref}(A)\) shown below.
\[A=\begin{bmatrix}1&2&1&0&-1\\1&2&2&3&0\\0&0&1&3&1\end{bmatrix}\rightsquigarrow \begin{bmatrix}1&2&0&-3&-2\\0&0&1&3&1\\0&0&0&0&0\end{bmatrix}=\mbox {rref}(A)\]
Find each of the following:
\[\text {rank}(A)=\answer {2}\]
\[\text {dim}\left (\text {row}(A)\right )=\answer {2}\]
\[\text {dim}\left (\text {col}(A)\right )=\answer {2}\]
\[\text {dim}\left (\text {null}(A)\right )=\answer {3}\]

Select an appropriate basis for \(\text {row}(A)\).

\(\left \{ \begin{bmatrix}1&2&1&0&-1\end{bmatrix}, \begin{bmatrix}1&2&2&3&0\end{bmatrix}, \begin{bmatrix}0&0&1&3&1\end{bmatrix}\right \}\) \(\left \{ \begin{bmatrix}1&2&0&-3&-2\end{bmatrix}, \begin{bmatrix}0&0&1&3&1\end{bmatrix}, \begin{bmatrix}0&0&0&0&0\end{bmatrix}\right \}\) \(\left \{ \begin{bmatrix}1&2&0&-3&-2\end{bmatrix}, \begin{bmatrix}0&0&1&3&1\end{bmatrix}\right \}\)

Using the algorithm presented in this text, find a basis for \(\text {col}(A)\).

\[\left \{\begin{bmatrix}\answer {1}\\\answer {1}\\\answer {0}\end{bmatrix}, \begin{bmatrix}\answer {1}\\\answer {2}\\\answer {1}\end{bmatrix}\right \}\]
Let \(\mathcal {B}=\left \{\vec {v}_1, \vec {v}_2\right \}\) be a basis for the plane depicted below. Find the coordinate vector for \(\vec {x}\) with respect to \(\mathcal {B}\).

[Picture]

\[\begin{bmatrix}\answer {1}\\\answer {2}\end{bmatrix}\]
Suppose that \(\mathcal {B}_1=\left \{\vec {v}_1, \vec {v}_2\right \}\) is an ordered basis for some subspace \(V\) of \(\RR ^n\). Let \(\mathcal {B}_2=\left \{-\vec {v}_2, -2\vec {v}_1\right \}\). Verify that \(\mathcal {B}_2\) is also an ordered basis for \(V\).

Let \(\vec {w}\) be a vector in \(V\). If the coordinate vector for \(\vec {w}\) with respect to \(\mathcal {B}_1\) is \(\begin{bmatrix}2\\-1\end{bmatrix}\), find the coordinate vector for \(\vec {w}\) with respect to \(\mathcal {B}_2\).

\[\begin{bmatrix}\answer {1}\\\answer {-1}\end{bmatrix}\]

Determine whether each set, \(S\), of vectors is closed under vector addition, and scalar multiplication.

Let \(S\) be the set of all vectors contained in a sphere of radius 1 centered at the origin.

[Picture]

\(S\) is closed under scalar multiplication. \(S\) is closed under vector addition. None of the above.

Let \(S\) be the set of all vectors contained in an infinite double cone.

[Picture]

\(S\) is closed under scalar multiplication. \(S\) is closed under vector addition. None of the above.
Find the dimension of \(V\) if
\[V=\text {span}\left (\begin{bmatrix}1\\2\\-1\\1\end{bmatrix},\begin{bmatrix}0\\2\\1\\-1\end{bmatrix}, \begin{bmatrix}2\\2\\-3\\3\end{bmatrix}, \begin{bmatrix}-1\\-2\\1\\-1\end{bmatrix}\right ) \]
\[\text {dim}(V)=\answer {2}\]

True or False? If False, you should come up with a counterexample. If True, can you give a proof?

1.
If \(V\) is a subspace of \(\RR ^n\) and \(\vec {x}+\vec {y}\) is in \(V\), then \(\vec {x}\) is in \(V\) or \(\vec {y}\) is in \(V\).
True False
2.
If \(V\) is a set in \(\RR ^n\) such that \(c_1{\vec {v}_1}+c_2{\vec {v}_2}\) is in \(V\) whenever \(\vec {v}_1\) and \(\vec {v}_2\) are in \(V\) for any scalars \(c_1\), \(c_2\), then \(V\) is a subspace.
True False
3.
Every set of four non-zero vectors in \(\RR ^4\) is a basis.
True False
4.
\(\RR ^3\) has a basis of the form \(\left \{\vec {x},\vec {x}+\vec {y},\vec {y}\right \}\).
True False
Suppose a linear transformation \(T:\RR ^2\rightarrow \RR ^2\) is such that
\[T\left (\begin{bmatrix}2\\-3\end{bmatrix}\right )=\begin{bmatrix}1\\0\end{bmatrix}\]
\[T\left (\begin{bmatrix}-1\\7\end{bmatrix}\right )=\begin{bmatrix}2\\1\end{bmatrix}\]
Then,
\[T\left (\begin{bmatrix}1\\4\end{bmatrix}\right )=\begin{bmatrix}\answer {3}\\\answer {1}\end{bmatrix}\]
Find the standard matrix \(M\) of a linear transformation \(T_M:\RR ^3\rightarrow \RR ^2\) if
\[T\left (\begin{bmatrix}1\\1\\0\end{bmatrix}\right )=\begin{bmatrix}3\\0\end{bmatrix};\quad T\left (\begin{bmatrix}0\\1\\1\end{bmatrix}\right )=\begin{bmatrix}1\\4\end{bmatrix};\quad T\left (\begin{bmatrix}1\\0\\1\end{bmatrix}\right )=\begin{bmatrix}2\\2\end{bmatrix}\]
\[M=\begin{bmatrix}\answer {2} & \answer {1} & \answer {0}\\\answer {-1} &\answer {1} & \answer {3}\end{bmatrix}\]
Suppose that an invertible linear transformation \(T:\RR ^2\rightarrow \RR ^2\) is such that
\[T\left (\begin{bmatrix}3\\-1\end{bmatrix}\right )=\begin{bmatrix}4\\-7\end{bmatrix};\quad T\left (\begin{bmatrix}2\\1\end{bmatrix}\right )=\begin{bmatrix}-5\\4\end{bmatrix}\]

Find the vector whose image under \(T\) is \(\begin{bmatrix}3\\-10\end{bmatrix}\)

\[T\left (\begin{bmatrix}\answer {8}\\\answer {-1}\end{bmatrix}\right )=\begin{bmatrix}3\\-10\end{bmatrix}\]

True or False? If False, you should come up with a counterexample. If True, can you give a proof?

1.
\(T : \RR ^2 \to \RR ^2\), given by \(T(x, y) = (x, -y)\), is a linear transformation.
True False
2.
\(T : \RR ^n \to \RR \), given by \(T(\vec {x}) = \vec {x} \cdot \vec {z}\) for some fixed vector \(\vec {z} \in \RR ^n\), is a linear transformation.
True False
3.
\(T : \RR \to \RR \), given by \(T(x) = x^2\), is a linear transformation.
True False
4.
Let \(T : \RR ^n \to \RR ^m\) be a linear transformation and let \(\vec {v}_{1}, \dots , \vec {v}_{k}\) denote vectors in \(\RR ^n\). If \(\{T(\vec {v}_{1}), \dots , T(\vec {v}_{k})\}\) is linearly independent, then \(\{\vec {v}_{1}, \dots , \vec {v}_{k}\}\) is also linearly independent.
True False
5.
Let \(T : \RR ^2 \to \RR ^2\) be a linear transformation and suppose \(\vec {v}_{1}, \vec {v}_{2}\) denote vectors in \(\RR ^2\). If \(\{\vec {v}_{1}, \vec {v}_{2}\}\) is linearly independent, then \(\{T(\vec {v}_{1}), T(\vec {v}_{2})\}\) is also linearly independent.
True False
Let
\[A=\begin{bmatrix}2 & 1 & -1 & 3\\1 & 0 & 3 & 1\\1 & 1 & -4 & 2\end{bmatrix}\]

Use techniques discussed in Image and Kernel of a Linear Transformation to find the basis for the kernel and the image of the linear transformation, \(T_A\), induced by \(A\).

Basis for \(\mbox {im}(T_A)\): \(\left \{\begin{bmatrix}\answer {2}\\\answer {1}\\\answer {1}\end{bmatrix}, \begin{bmatrix}\answer {1}\\\answer {0}\\\answer {1}\end{bmatrix}\right \}\)

Basis for \(\mbox {ker}(T_A)\): \(\left \{\begin{bmatrix}\answer {-3}\\\answer {7}\\1\\0\end{bmatrix}, \begin{bmatrix}\answer {-1}\\\answer {-1}\\0\\1\end{bmatrix}\right \}\)