How to Find the Determinant of the $3\times 3$ Matrix
Problem 138
Find the determinant of the matix
\[A=\begin{bmatrix}
100 & 101 & 102 \\
101 &102 &103 \\
102 & 103 & 104
\end{bmatrix}.\]
Add to solve later Add to solve later 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 $U$.
Add to solve later 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 later 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 $\rk(AB)=\rk(A)$.
Add to solve later 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) $T=\{f(x) \in V \mid f(0)=f(1)+3\}$.
Add to solve later 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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Add to solve later 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.)
Add to solve later 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.
Add to solve later 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.
Add to solve later 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 $G$.
Add to solve later 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.
Add to solve later 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 $N_2$.
Add to solve later 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.
Add to solve later 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 $\R^3$.
Add to solve later 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 $p^2$.
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Add to solve later 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.
Add to solve later Any Finite Group Has a Composition Series
Problem 122
Let $G$ be a finite group. Then show that $G$ has a composition series.
Add to solve later The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$
Problem 121
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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Add to solve later If Vectors are Linearly Dependent, then What Happens When We Add One More Vectors?
Problem 120
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 later Subset of Vectors Perpendicular to Two Vectors is a Subspace
Problem 119
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 $\R^3$.
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