Author: Yu

10/13/2016

Find a Value of a Linear Transformation From $\R^2$ to $\R^3$

Problem 142

Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix}
1 \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1
\end{bmatrix}$ are unit vectors of $\R^2$ and
\[\mathbf{u}_1= \begin{bmatrix}
-1 \\
0 \\
1
\end{bmatrix}, \quad \mathbf{u}_2=\begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}.\] Then find $T\left(\begin{bmatrix}
3 \\
-2
\end{bmatrix}\right)$.

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10/11/2016

Linear Independent Vectors and the Vector Space Spanned By Them

Problem 141

Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$.
Let $\mathbf{u}_{n+1}\in V$. Show that $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n, \mathbf{u}_{n+1}$ are linearly independent if and only if $\mathbf{u}_{n+1} \not \in U$.

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10/10/2016

Rank and Nullity of a Matrix, Nullity of Transpose

Problem 140

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

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10/09/2016

Sum of Squares of Hermitian Matrices is Zero, then Hermitian Matrices Are All Zero

Problem 139

Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if
\[A_1^2+A_2^2+\cdots+A_m^2=\calO,\] where $\calO$ is the $n \times n$ zero matrix, then we have $A_i=\calO$ for each $i=1,2, \dots, m$.

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10/09/2016

How to Find the Determinant of the $3\times 3$ Matrix

Problem 138

Find the determinant of the matix
\[A=\begin{bmatrix}
100 & 101 & 102 \\
101 &102 &103 \\
102 & 103 & 104
\end{bmatrix}.\]

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10/08/2016

Find a Basis and Determine the Dimension of a Subspace of All Polynomials of Degree $n$ or Less

Problem 137

Let $P_n(\R)$ be the vector space over $\R$ consisting of all degree $n$ or less real coefficient polynomials. Let
\[U=\{ p(x) \in P_n(\R) \mid p(1)=0\}\] be a subspace of $P_n(\R)$.

Find a basis for $U$ and determine the dimension of $U$.

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10/07/2016

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 $A^{\trans}$.

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10/06/2016

Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$

Problem 135

Let $A$ 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 $\rk(AB)=\rk(A)$.

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10/05/2016

Subspaces of the Vector Space of All Real Valued Function on the Interval

Problem 134

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

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10/04/2016

Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?

Problem 133

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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10/03/2016

Find a Basis For the Null Space of a Given $2\times 3$ Matrix

Problem 132

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.)

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10/02/2016

Find a Basis and the Dimension of the Subspace of the 4-Dimensional Vector Space

Problem 131

Let $V$ 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.

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10/01/2016

Multiplicative Groups of Real Numbers and Complex Numbers are not Isomorphic

Problem 130

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.

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10/01/2016

Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup

Problem 129

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 $G$.

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09/30/2016

If a Subgroup $H$ is in the Center of a Group $G$ and $G/H$ is Nilpotent, then $G$ is Nilpotent

Problem 128

Let $G$ 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.

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09/29/2016

Normal Subgroups, Isomorphic Quotients, But Not Isomorphic

Problem 127

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 $N_2$.

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09/28/2016

Find Values of $a$ so that the Matrix is Nonsingular

Problem 126

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.

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09/28/2016

Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$

Problem 125

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 $\R^3$.

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09/27/2016

The Index of the Center of a Non-Abelian $p$-Group is Divisible by $p^2$

Problem 124

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 $p^2$.
Read solution

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09/27/2016

Infinite Cyclic Groups Do Not Have Composition Series

Problem 123

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

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