## Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)

## Problem 136

Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.

Add to solve laterLet $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.

Add to solve laterLet $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings.

**(a)** $\rk(AB) \leq \rk(A)$.

**(b)** If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.

Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not.

**(a)** $S=\{f(x) \in V \mid f(0)=f(1)\}$.

**(b)** $T=\{f(x) \in V \mid f(0)=f(1)+3\}$.

Let

\[A=\begin{bmatrix}

1 & 1 & 0 \\

1 &1 &0

\end{bmatrix}\]
be a matrix.

Find a basis of the null space of the matrix $A$.

(Remark: a null space is also called a kernel.)

Add to solve laterLet $V$ be the following subspace of the $4$-dimensional vector space $\R^4$.

\[V:=\left\{ \quad\begin{bmatrix}

x_1 \\

x_2 \\

x_3 \\

x_4

\end{bmatrix} \in \R^4

\quad \middle| \quad

x_1-x_2+x_3-x_4=0 \quad\right\}.\]
Find a basis of the subspace $V$ and its dimension.

Let $S$ be the following subset of the 3-dimensional vector space $\R^3$.

\[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix}

x_1 \\

x_2 \\

x_3

\end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \]
where $\Z$ is the set of all integers.

Determine whether $S$ is a subspace of $\R^3$.

Let $A$ be an $m \times n$ real matrix. Then the **null space** $\calN(A)$ of $A$ is defined by

\[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\]
That is, the null space is the set of solutions to the homogeneous system $A\mathbf{x}=\mathbf{0}_m$.

Prove that the null space $\calN(A)$ is a subspace of the vector space $\R^n$.

(Note that the null space is also called the **kernel** of $A$.)

Read solution

Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by

\[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\]

Prove that the subset $W$ is a subspace of $\R^3$.

Read solution

Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.

That is,

\begin{equation*}

V:=\left\{ A=\begin{bmatrix}

a_{11} & 0 & \dots & 0 \\

0 &a_{22} & \dots & 0 \\

0 & 0 & \ddots & \vdots \\

0 & 0 & \dots & a_{nn}

\end{bmatrix} \quad \middle| \quad

\begin{array}{l}

a_{11}, \dots, a_{nn} \in \C,\\

\tr(A)=0 \\

\end{array}

\right\}

\end{equation*}

Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.

**(a)** Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)

**(b)** Show that matrices

\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\]
are a basis for the vector space $V$.

**(c)** Find the dimension of $V$.

Read solution

Let $V$ and $W$ be subspaces of $\R^n$ such that $V \cap W =\{\mathbf{0}\}$ and $\dim(V)+\dim(W)=n$.

**(a)** If $\mathbf{v}+\mathbf{w}=\mathbf{0}$, where $\mathbf{v}\in V$ and $\mathbf{w}\in W$, then show that $\mathbf{v}=\mathbf{0}$ and $\mathbf{w}=\mathbf{0}$.

**(b)** If $B_1$ is a basis for the subspace $V$ and $B_2$ is a basis for the subspace $W$, then show that the union $B_1\cup B_2$ is a basis for $R^n$.

**(c)** If $\mathbf{x}$ is in $\R^n$, then show that $\mathbf{x}$ can be written in the form $\mathbf{x}=\mathbf{v}+\mathbf{w}$, where $\mathbf{v}\in V$ and $\mathbf{w} \in W$.

**(d)** Show that the representation obtained in part (c) is unique.

Let $T: \R^3 \to \R^3$ be the linear transformation given by orthogonal projection to the line spanned by $\begin{bmatrix}

1 \\

2 \\

2

\end{bmatrix}$.

**(a)** Find a formula for $T(\mathbf{x})$ for $\mathbf{x}\in \R^3$.

**(b)** Find a basis for the image subspace of $T$.

**(c)** Find a basis for the kernel subspace of $T$.

**(d)** Find the $3 \times 3$ matrix for $T$ with respect to the standard basis for $\R^3$.

**(e)** Find a basis for the orthogonal complement of the kernel of $T$. (The orthogonal complement is the subspace of all vectors perpendicular to a given subspace, in this case, the kernel.)

**(f)** Find a basis for the orthogonal complement of the image of $T$.

**(g)** What is the rank of $T$?

(*Johns Hopkins University Exam*)

Find a basis for the subspace $W$ of all vectors in $\R^4$ which are perpendicular to the columns of the matrix

\[A=\begin{bmatrix}

11 & 12 & 13 & 14 \\

21 &22 & 23 & 24 \\

31 & 32 & 33 & 34 \\

41 & 42 & 43 & 44

\end{bmatrix}.\]

(*Harvard University Exam*)

Suppose that $S$ is a fixed invertible $3$ by $3$ matrix. This question is about all the matrices $A$ that are diagonalized by $S$, so that $S^{-1}AS$ is diagonal. Show that these matrices $A$ form a subspace of $3$ by $3$ matrix space.

(*MIT-Massachusetts Institute of Technology Exam*)

Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent?

**(a)** At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ for $i=1,2,3,4$.

**(b)** At $0$ each of the polynomials has the value $1$. Namely $p_i(0)=1$ for $i=1,2,3,4$.

(*University of California, Berkeley*)