Determine bases for $\calN(A)$ and $\calN(A^{T}A)$ when
\[
A=
\begin{bmatrix}
1 & 2 & 1 \\
1 & 1 & 3 \\
0 & 0 & 0
\end{bmatrix}
.
\]
Then, determine the ranks and nullities of the matrices $A$ and $A^{\trans}A$.
Add to solve later Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where
\[
\mathbf{v}_{1}=
\begin{bmatrix}
1 \\ 2 \\ 2 \\ -1
\end{bmatrix}
,\;\mathbf{v}_{2}=
\begin{bmatrix}
1 \\ 3 \\ 1 \\ 1
\end{bmatrix}
,\;\mathbf{v}_{3}=
\begin{bmatrix}
1 \\ 5 \\ -1 \\ 5
\end{bmatrix}
,\;\mathbf{v}_{4}=
\begin{bmatrix}
1 \\ 1 \\ 4 \\ -1
\end{bmatrix}
,\;\mathbf{v}_{5}=
\begin{bmatrix}
2 \\ 7 \\ 0 \\ 2
\end{bmatrix}
.\]
Find a basis for the span $\Span(S)$.
Add to solve later Let $\mathrm{P}_3$ denote the set of polynomials of degree $3$ or less with real coefficients. Consider the ordered basis
\[B = \left\{ 1+x , 1+x^2 , x – x^2 + 2x^3 , 1 – x – x^2 \right\}.\]
Write the coordinate vector for the polynomial $f(x) = -3 + 2x^3$ in terms of the basis $B$.
Add to solve later For an integer $n > 0$, let $\mathrm{P}_n$ denote the vector space of polynomials with real coefficients of degree $2$ or less. Define the map $T : \mathrm{P}_2 \rightarrow \mathrm{P}_4$ by
\[ T(f)(x) = f(x^2).\]
Determine if $T$ is a linear transformation.
If it is, find the matrix representation for $T$ relative to the basis $\mathcal{B} = \{ 1 , x , x^2 \}$ of $\mathrm{P}_2$ and $\mathcal{C} = \{ 1 , x , x^2 , x^3 , x^4 \}$ of $\mathrm{P}_4$.
Add to solve later Let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.
Let $T : \mathrm{P}_3 \rightarrow \mathrm{P}_{5}$ be the map defined by, for $f \in \mathrm{P}_3$,
\[T (f) (x) = ( x^2 – 2) f(x).\]
Determine if $T(x)$ is a linear transformation. If it is, find the matrix representation of $T$ relative to the standard basis of $\mathrm{P}_3$ and $\mathrm{P}_{5}$.
Add to solve later For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.
Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, for $f \in \mathrm{P}_n$,
\[T (f) (x) = x f(x).\]
Prove that $T$ is a linear transformation, and find its range and nullspace.
Add to solve later Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis for $\R^2$. Let $S:=[\mathbf{v}_1, \mathbf{v}_2]$.
Note that as the column vectors of $S$ are linearly independent, the matrix $S$ is invertible.
Prove that for each vector $\mathbf{v} \in V$, the vector $S^{-1}\mathbf{v}$ is the coordinate vector of $\mathbf{v}$ with respect to the basis $B$.
Add to solve later Let $A$ and $B$ be $n\times n$ matrices and assume that they commute: $AB=BA$.
Then prove that the matrices $A$ and $B$ share at least one common eigenvector.
Add to solve later Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less.
Let
\[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\]
where
\begin{align*}
p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\
p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3.
\end{align*}
(a) Find a basis $Q$ of the span $\Span(S)$ consisting of polynomials in $S$.
(b) For each polynomial in $S$ that is not in $Q$, find the coordinate vector with respect to the basis $Q$.
(The Ohio State University, Linear Algebra Midterm)
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Add to solve later Let $T:\R^2 \to \R^3$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
3 \\
2
\end{bmatrix} \,\right)
=\begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix} \text{ and }
T\left(\, \begin{bmatrix}
4\\
3
\end{bmatrix} \,\right)
=\begin{bmatrix}
0 \\
-5 \\
1
\end{bmatrix}.\]
(a) Find the matrix representation of $T$ (with respect to the standard basis for $\R^2$).
(b) Determine the rank and nullity of $T$.
(The Ohio State University, Linear Algebra Midterm)
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Add to solve later Let
\[A=\begin{bmatrix}
1 & -1 & 0 & 0 \\
0 &1 & 1 & 1 \\
1 & -1 & 0 & 0 \\
0 & 2 & 2 & 2\\
0 & 0 & 0 & 0
\end{bmatrix}.\]
(a) Find a basis for the null space $\calN(A)$.
(b) Find a basis of the range $\calR(A)$.
(c) Find a basis of the row space for $A$.
(The Ohio State University, Linear Algebra Midterm)
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Add to solve later Let $V$ be the vector space over $\R$ of all real $2\times 2$ matrices.
Let $W$ be the subset of $V$ consisting of all symmetric matrices.
(a) Prove that $W$ is a subspace of $V$.
(b) Find a basis of $W$.
(c) Determine the dimension of $W$.
Add to solve later Let $V$ be a subset of $\R^4$ consisting of vectors that are perpendicular to vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, where
\[\mathbf{a}=\begin{bmatrix}
1 \\
0 \\
1 \\
0
\end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix}
1 \\
1 \\
0 \\
0
\end{bmatrix}, \quad \mathbf{c}=\begin{bmatrix}
0 \\
1 \\
-1 \\
0
\end{bmatrix}.\]
Namely,
\[V=\{\mathbf{x}\in \R^4 \mid \mathbf{a}^{\trans}\mathbf{x}=0, \mathbf{b}^{\trans}\mathbf{x}=0, \text{ and } \mathbf{c}^{\trans}\mathbf{x}=0\}.\]
(a) Prove that $V$ is a subspace of $\R^4$.
(b) Find a basis of $V$.
(c) Determine the dimension of $V$.
Add to solve later Let $V$ be a subspace of $\R^n$.
Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a basis of the subspace $V$.
Prove that every basis of $V$ consists of $k$ vectors in $V$.
Add to solve later Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$.
(a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$.
(b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of $\R^3$.
Add to solve later Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \,\right)=\begin{bmatrix}
x_1+3x_2-2x_3 \\
2x_1+3x_2 \\
x_2-x_3
\end{bmatrix}.\]
Determine whether $T$ is an isomorphism and if so find the formula for the inverse linear transformation $T^{-1}$.
Add to solve later Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$.
Consider a linear transformation $T:U\to V$.
Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).
Add to solve later Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less.
Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where
\begin{align*}
p_1(x)&=-1+x+2x^2, \quad p_2(x)=x+3x^2\\
p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2.
\end{align*}
(a) Find a basis of $P_2$ among the vectors of $S$. (Explain why it is a basis of $P_2$.)
(b) Let $B’$ be the basis you obtained in part (a).
For each vector of $S$ which is not in $B’$, find the coordinate vector of it with respect to the basis $B’$.
(The Ohio State University, Linear Algebra Final Exam Problem)
Add to solve later Let $T:\R^2 \to \R^3$ be a linear transformation given by
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \,\right)
=
\begin{bmatrix}
x_1-x_2 \\
x_2 \\
x_1+ x_2
\end{bmatrix}.\]
Find an orthonormal basis of the range of $T$.
(The Ohio State University, Linear Algebra Final Exam Problem)
Add to solve later Let
\[A=\begin{bmatrix}
1 & 2 & 1 \\
-1 &4 &1 \\
2 & -4 & 0
\end{bmatrix}.\]
The matrix $A$ has an eigenvalue $2$.
Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$.
(The Ohio State University, Linear Algebra Final Exam Problem)
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