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by Yu · Published 09/13/2017 · Last modified 01/16/2018
Determine a Condition on $a, b$ so that Vectors are Linearly Dependent
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
a \\
\end{bmatrix}\] be vectors in $\R^3$.
Determine a condition on the scalars $a, b$ so that the set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent.
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Two Matrices are Nonsingular if and only if the Product is Nonsingular
An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$.
Using the definition of a nonsingular matrix, prove the following statements.
(a) If $A$ and $B$ are $n\times n$ nonsingular matrix, then the product $AB$ is also nonsingular.
(b) Let $A$ and $B$ be $n\times n$ matrices and suppose that the product $AB$ is nonsingular. Then:
The matrix $B$ is nonsingular.
The matrix $A$ is nonsingular. (You may use the fact that a nonsingular matrix is invertible.)
A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors
Let $A$ be a singular $n\times n$ matrix.
\[\mathbf{e}_1=\begin{bmatrix}
\vdots \\
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
\end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix}
\end{bmatrix}\] be unit vectors in $\R^n$.
Prove that at least one of the following matrix equations
\[A\mathbf{x}=\mathbf{e}_i\] for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$.
Click here if solved 9
by Yu · Published 09/10/2017
The Matrix $[A_1, \dots, A_{n-1}, A\mathbf{b}]$ is Always Singular, Where $A=[A_1,\dots, A_{n-1}]$ and $\mathbf{b}\in \R^{n-1}$.
Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector.
Then the product $A\mathbf{b}$ is an $n$-dimensional vector.
Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$.
Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$.
Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$
For each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$.
(a) $A=\begin{bmatrix}
4 & 2\\
2& 1
\end{bmatrix}$.
(b) $A=\begin{bmatrix}
The Transpose of a Nonsingular Matrix is Nonsingular
Let $A$ be an $n\times n$ nonsingular matrix.
Prove that the transpose matrix $A^{\trans}$ is also nonsingular.
If the Quotient is an Infinite Cyclic Group, then Exists a Normal Subgroup of Index $n$
Let $N$ be a normal subgroup of a group $G$.
Suppose that $G/N$ is an infinite cyclic group.
Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.
Construction of a Symmetric Matrix whose Inverse Matrix is Itself
Let $\mathbf{v}$ be a nonzero vector in $\R^n$.
Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$.
Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by
\[A=I-a\mathbf{v}\mathbf{v}^{\trans},\] where $I$ is the $n\times n$ identity matrix.
Prove that $A$ is a symmetric matrix and $AA=I$.
Conclude that the inverse matrix is $A^{-1}=A$.
The Range and Null Space of the Zero Transformation of Vector Spaces
Let $U$ and $V$ be vector spaces over a scalar field $\F$.
Define the map $T:U\to V$ by $T(\mathbf{u})=\mathbf{0}_V$ for each vector $\mathbf{u}\in U$.
(a) Prove that $T:U\to V$ is a linear transformation.
(Hence, $T$ is called the zero transformation.)
(b) Determine the null space $\calN(T)$ and the range $\calR(T)$ of $T$.
If Generators $x, y$ Satisfy the Relation $xy^2=y^3x$, $yx^2=x^3y$, then the Group is Trivial
Let $x, y$ be generators of a group $G$ with relation
xy^2=y^3x,\tag{1}\\
yx^2=x^3y.\tag{2}
Prove that $G$ is the trivial group.
Find the Inverse Linear Transformation if the Linear Transformation is an Isomorphism
Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula
\[T\left(\, \begin{bmatrix}
x_1 \\
x_3
\end{bmatrix} \,\right)=\begin{bmatrix}
x_1+3x_2-2x_3 \\
2x_1+3x_2 \\
x_2+x_3
\end{bmatrix}.\]
Determine whether $T$ is an isomorphism and if so find the formula for the inverse linear transformation $T^{-1}$.
Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.
1 & 3 & -2 \\
2 &3 &0 \\
0 & 1 & -1
\end{bmatrix}$
1 & 0 & 2 \\
-1 &-3 &2 \\
Click here if solved 112
The Sum of Cosine Squared in an Inner Product Space
Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$.
Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$.
Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$.
Prove that
\[\cos ^2\theta_1+\cdots+\cos^2 \theta_n=1.\]
Rotation Matrix in the Plane and its Eigenvalues and Eigenvectors
Consider the $2\times 2$ matrix
\[A=\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta& \cos \theta \end{bmatrix},\] where $\theta$ is a real number $0\leq \theta < 2\pi$.
(a) Find the characteristic polynomial of the matrix $A$.
(b) Find the eigenvalues of the matrix $A$.
(c) Determine the eigenvectors corresponding to each of the eigenvalues of $A$.
Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent
By calculating the Wronskian, determine whether the set of exponential functions
\[\{e^x, e^{2x}, e^{3x}\}\] is linearly independent on the interval $[-1, 1]$.
A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$
An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that
$AB=I$, and
$BA=I$,
where $I$ is the $n\times n$ identity matrix.
If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted by $A^{-1}$.
In this problem, we prove that if $B$ satisfies the first condition, then it automatically satisfies the second condition.
So if we know $AB=I$, then we can conclude that $B=A^{-1}$.
Let $A$ and $B$ be $n\times n$ matrices.
Suppose that we have $AB=I$, where $I$ is the $n \times n$ identity matrix.
Prove that $BA=I$, and hence $A^{-1}=B$.
Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$
Let $A$ be an $n\times n$ nonsingular matrix with integer entries.
Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$.
Find Inverse Matrices Using Adjoint Matrices
Let $A$ be an $n\times n$ matrix.
The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be
\[C_{ij}=(-1)^{ij}\det(M_{ij}),\] where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column.
Then consider the $n\times n$ matrix $C=(C_{ij})$, and define the $n\times n$ matrix $\Adj(A)=C^{\trans}$.
The matrix $\Adj(A)$ is called the adjoint matrix of $A$.
When $A$ is invertible, then its inverse can be obtained by the formula
\[A^{-1}=\frac{1}{\det(A)}\Adj(A).\]
For each of the following matrices, determine whether it is invertible, and if so, then find the invertible matrix using the above formula.
0 &-1 &2 \\
0 & 0 & 1
(b) $B=\begin{bmatrix}
Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$
Let $V$ be a vector space over the field of real numbers $\R$.
Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$.
The Product of Distinct Sylow $p$-Subgroups Can Never be a Subgroup
Let $G$ a finite group and let $H$ and $K$ be two distinct Sylow $p$-group, where $p$ is a prime number dividing the order $|G|$ of $G$.
Prove that the product $HK$ can never be a subgroup of the group $G$.
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