## Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2

## Problem 646

**(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.

of the day

**(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.

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)** $E = \begin{bmatrix} -2 & 3 & 1 \end{bmatrix}$.

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)*

Read solution

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 give 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*)

Read solution

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*)

Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.

Add to solve laterLet $A$ 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*)

**(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 $\lambda=0$.

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*)

Read solution

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*)

Read solution

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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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 $\mathbf{v}\in \calN(T)$.

**(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).

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.

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.

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 $C$.

Read solution 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*)

Read solution

Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix}

x_1 \\

x_2

\end{bmatrix}\right )=\begin{bmatrix}

x_1-x_2 \\

x_1+x_2 \\

x_2

\end{bmatrix}$.

**(a) **Show that $T$ is a linear transformation.

**(b)** Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.

**(c)** Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.

Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$.

The dimension of the nullspace of $A$ is called the nullity of $A$.

Prove the followings.

**(a)** $\calN(A)=\calN(A^{\trans}A)$.

**(b)** $\rk(A)=\rk(A^{\trans}A)$.