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

Find **a square root** of the matrix

\[A=\begin{bmatrix}

1 & 3 & -3 \\

0 &4 &5 \\

0 & 0 & 9

\end{bmatrix}.\]

How many square roots does this matrix have?

(*University of California, Berkeley Qualifying Exam*)

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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 $\R^{\times}=\R\setminus \{0\}$ be the multiplicative group of real numbers.

Let $\C^{\times}=\C\setminus \{0\}$ be the multiplicative group of complex numbers.

Then show that $\R^{\times}$ and $\C^{\times}$ are not isomorphic as groups.

Let $G$ be a group and $H$ and $K$ be subgroups of $G$.

For $h \in H$, and $k \in K$, we define the commutator $[h, k]:=hkh^{-1}k^{-1}$.

Let $[H,K]$ be a subgroup of $G$ generated by all such commutators.

Show that if $H$ and $K$ are normal subgroups of $G$, then the subgroup $[H, K]$ is normal in $G$.

Add to solve laterLet $G$ be a nilpotent group and let $H$ be a subgroup such that $H$ is a subgroup in the center $Z(G)$ of $G$.

Suppose that the quotient $G/H$ is nilpotent.

Then show that $G$ is also nilpotent.

Add to solve later Let $G$ be a group. Suppose that $H_1, H_2, N_1, N_2$ are all normal subgroup of $G$, $H_1 \lhd N_2$, and $H_2 \lhd N_2$.

Suppose also that $N_1/H_1$ is isomorphic to $N_2/H_2$. Then prove or disprove that $N_1$ is isomorphic to $N_2$.

Let $A$ be the following $3 \times 3$ matrix.

\[A=\begin{bmatrix}

1 & 1 & -1 \\

0 &1 &2 \\

1 & 1 & a

\end{bmatrix}.\]
Determine the values of $a$ so that the matrix $A$ is nonsingular.

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 $p$ be a prime number.

Let $G$ be a non-abelian $p$-group.

Show that the index of the center of $G$ is divisible by $p^2$.

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Let $G$ be an infinite cyclic group. Then show that $G$ does not have a composition series.

Add to solve laterLet $G$ be a finite group. Then show that $G$ has a composition series.

Add to solve later 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 $A$.)

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Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$ are linearly dependent $n$-dimensional real vectors.

For any vector $\mathbf{v}_{r+1} \in \R^n$, determine whether the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly independent or linearly dependent.

Add to solve laterLet $\mathbf{a}$ 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 $\R^3$.

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Let $G$ be a finite group of order $18$.

Show that the group $G$ is solvable.

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Let $G$ be a finite group and $P$ be a nontrivial Sylow subgroup of $G$.

Let $H$ be a subgroup of $G$ containing the normalizer $N_G(P)$ of $P$ in $G$.

Then show that $N_G(H)=H$.

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