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 $A$ be an $m \times n$ matrix.
Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$.
Add to solve later (a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.
(b) Find all such matrices with rank 2.
Add to solve later For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix.
(a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$.
(b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$.
(c) $C = \begin{bmatrix} 2 & -2 & 4 \\ 4 & 1 & -2 \\ 6 & -1 & 2 \end{bmatrix}$.
(d) $D = \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}$.
(e) $E = \begin{bmatrix} -2 & 3 & 1 \end{bmatrix}$.
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 $U$ 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 $A$ be a square matrix and its characteristic polynomial is given by
\[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\]
Find the rank of $A$.
(The Ohio State University, Linear Algebra Final Exam Problem)
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Add to solve later Let $\mathbf{u}=\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation
\[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\]
(a) Calculate the null space $\calN(T)$, a basis for $\calN(T)$ and nullity of $T$.
(b) Only by using part (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard basis of $\R^3$.
(c) Calculate the range $\calR(T)$, a basis for $\calR(T)$ and the rank of $T$.
(d) Calculate the matrix $A$ representing $T$ with respect to the standard basis for $\R^3$.
(e) Let
\[B=\left\{\, \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \begin{bmatrix}
-1 \\
1 \\
0
\end{bmatrix}, \begin{bmatrix}
0 \\
-1 \\
1
\end{bmatrix} \,\right\}\]
be a basis for $\R^3$.
Calculate the coordinates of $\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}$ with respect to $B$.
(The Ohio State University, Linear Algebra Exam Problem)
Add to solve later Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.
Add to solve later Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities.
What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix?
(The Ohio State University, Linear Algebra Final Exam Problem)
Add to solve later (a) Let
\[A=\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 &1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1
\end{bmatrix}.\]
Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue.
(b) Let
\[A=\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 &1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1
\end{bmatrix}.\]
One of the eigenvalues of the matrix $A$ is $\lambda=0$. Find the geometric multiplicity of the eigenvalue $\lambda=0$.
Add to solve later Let $T: \R^2 \to \R^2$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
1 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
4 \\
1
\end{bmatrix}, T\left(\, \begin{bmatrix}
0 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
3 \\
2
\end{bmatrix}.\]
Then find the matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for every $\mathbf{x}\in \R^2$, and find the rank and nullity of $T$.
(The Ohio State University, Linear Algebra Exam Problem)
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Add to solve later Let $T:\R^3 \to \R^2$ be a linear transformation such that
\[ T(\mathbf{e}_1)=\begin{bmatrix}
1 \\
0
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
0 \\
1
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
1 \\
0
\end{bmatrix},\]
where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard basis of $\R^3$.
Then find the rank and the nullity of $T$.
(The Ohio State University, Linear Algebra Exam Problem)
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Add to solve later Let $A=\begin{bmatrix}
1 & 0 & 1 \\
0 &1 &0
\end{bmatrix}$.
(a) Find an orthonormal basis of the null space of $A$.
(b) Find the rank of $A$.
(c) Find an orthonormal basis of the row space of $A$.
(The Ohio State University, Linear Algebra Exam Problem)
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Add to solve later Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation.
Prove the followings.
(a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$.
(b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the nullspace $\calN(T)$ of $T$.
Let $\mathbf{w}$ be the $n$-dimensional vector that is not in $\calN(T)$. Then
\[B’=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}, \mathbf{w}\}\]
is a basis of $\R^n$.
(c) Each vector $\mathbf{u}\in \R^n$ can be expressed as
\[\mathbf{u}=\mathbf{v}+\frac{T(\mathbf{u})}{T(\mathbf{w})}\mathbf{w}\]
for some vector $\mathbf{v}\in \calN(T)$.
Add to solve later (a) Let $A=\begin{bmatrix}
1 & 3 & 0 & 0 \\
1 &3 & 1 & 2 \\
1 & 3 & 1 & 2
\end{bmatrix}$.
Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.
(b) Find the rank and nullity of the matrix $A$ in part (a).
Add to solve later Let $V$ be the vector space of all $2\times 2$ real matrices and let $P_3$ be the vector space of all polynomials of degree $3$ or less with real coefficients.
Let $T: P_3 \to V$ be the linear transformation defined by
\[T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix}
a_0+a_2 & -a_0+a_3\\
a_1-a_2 & -a_1-a_3
\end{bmatrix}\]
for any polynomial $a_0+a_1x+a_2x^2+a_3 \in P_3$.
Find a basis for the range of $T$, $\calR(T)$, and determine the rank of $T$, $\rk(T)$, and the nullity of $T$, $\nullity(T)$.
Also, prove that $T$ is not injective.
Add to solve later Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$.
Then prove the following statements.
(a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number.
(b) The rank of $A$ is even.
Add to solve later Let $A, B, C$ are $2\times 2$ diagonalizable matrices.
The graphs of characteristic polynomials of $A, B, C$ are shown below. The red graph is for $A$, the blue one for $B$, and the green one for $C$.
From this information, determine the rank of the matrices $A, B,$ and $C$.
Graphs of characteristic polynomials
Add to solve later Let $T:\R^4 \to \R^3$ be a linear transformation defined by
\[ T\left (\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix} \,\right) = \begin{bmatrix}
x_1+2x_2+3x_3-x_4 \\
3x_1+5x_2+8x_3-2x_4 \\
x_1+x_2+2x_3
\end{bmatrix}.\]
(a) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.
(b) Find a basis for the null space of $T$.
(c) Find the rank of the linear transformation $T$.
(The Ohio State University Linear Algebra Exam Problem)
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