## Prove the Cauchy-Schwarz Inequality

## Problem 355

Let $\mathbf{a}, \mathbf{b}$ be vectors in $\R^n$.

Prove the **Cauchy-Schwarz inequality**:

\[|\mathbf{a}\cdot \mathbf{b}|\leq \|\mathbf{a}\|\,\|\mathbf{b}\|.\]

Let $\mathbf{a}, \mathbf{b}$ be vectors in $\R^n$.

Prove the **Cauchy-Schwarz inequality**:

\[|\mathbf{a}\cdot \mathbf{b}|\leq \|\mathbf{a}\|\,\|\mathbf{b}\|.\]

Let $G$ be a group. Let $a$ and $b$ be elements of $G$.

If the order of $a, b$ are $m, n$ respectively, then is it true that the order of the product $ab$ divides $mn$? If so give a proof. If not, give a counterexample.

Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying

\[T\left(\, \begin{bmatrix}

1 \\

2

\end{bmatrix}\,\right)=\begin{bmatrix}

3 \\

4 \\

5

\end{bmatrix} \text{ and } T\left(\, \begin{bmatrix}

0 \\

1

\end{bmatrix} \,\right)=\begin{bmatrix}

0 \\

0 \\

1

\end{bmatrix}.\]
Find a general formula for

\[T\left(\, \begin{bmatrix}

x_1 \\

x_2

\end{bmatrix} \,\right).\]

(*The Ohio State University, Linear Algebra Math 2568 Exam Problem*)

A **hyperplane ** in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors

\[\begin{bmatrix}

x_1 \\

x_2 \\

\vdots \\

x_n

\end{bmatrix}\in \R^n\]
satisfying the linear equation of the form

\[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\]
where $a_1, a_2, \dots, a_n$ (at least one of $a_1, a_2, \dots, a_n$ is nonzero) and $b$ are real numbers.

Here at least one of $a_1, a_2, \dots, a_n$ is nonzero.

Consider the hyperplane $P$ in $\R^n$ described by the linear equation

\[a_1x_1+a_2x_2+\cdots+a_nx_n=0,\]
where $a_1, a_2, \dots, a_n$ are some fixed real numbers and not all of these are zero.

(The constant term $b$ is zero.)

Then prove that the hyperplane $P$ is a subspace of $R^{n}$ of dimension $n-1$.

Add to solve laterLet $R$ be a commutative ring with unity.

Then show that every maximal ideal of $R$ is a prime ideal.

Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$.

Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of $S$.

\[[v_1]_B=\begin{bmatrix}

1 \\

0 \\

0 \\

0

\end{bmatrix}, [v_2]_B=\begin{bmatrix}

0 \\

1 \\

0 \\

0

\end{bmatrix}, [v_3]_B=\begin{bmatrix}

1 \\

1 \\

0 \\

0

\end{bmatrix}.\]

Let $V$ be the vector space of all $2\times 2$ real matrices.

Let $S=\{A_1, A_2, A_3, A_4\}$, where

\[A_1=\begin{bmatrix}

1 & 2\\

-1& 3

\end{bmatrix}, A_2=\begin{bmatrix}

0 & -1\\

1& 4

\end{bmatrix}, A_3=\begin{bmatrix}

-1 & 0\\

1& -10

\end{bmatrix}, A_4=\begin{bmatrix}

3 & 7\\

-2& 6

\end{bmatrix}.\]
Then find a basis for the span $\Span(S)$.

Let $A$ be an $n\times n$ complex matrix.

Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as

\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.

Let $C$ be the companion matrix of the polynomial $p(x)$ given by

\[C=\begin{bmatrix}

0 & 0 & \dots & 0 &-a_0 \\

1 & 0 & \dots & 0 & -a_1 \\

0 & 1 & \dots & 0 & -a_2 \\

\vdots & & \ddots & & \vdots \\

0 & 0 & \dots & 1 & -a_{n-1}

\end{bmatrix}=

[\mathbf{e}_2, \mathbf{e}_3, \dots, \mathbf{e}_n, -\mathbf{a}],\]
where $\mathbf{e}_i$ is the unit vector in $\C^n$ whose $i$-th entry is $1$ and zero elsewhere, and the vector $\mathbf{a}$ is defined by

\[\mathbf{a}=\begin{bmatrix}

a_0 \\

a_1 \\

\vdots \\

a_{n-1}

\end{bmatrix}.\]

Then prove that the following two statements are equivalent.

- There exists a vector $\mathbf{v}\in \C^n$ such that

\[\mathbf{v}, A\mathbf{v}, A^2\mathbf{v}, \dots, A^{n-1}\mathbf{v}\] form a basis of $\C^n$. - There exists an invertible matrix $S$ such that $S^{-1}AS=C$.

(Namely, $A$ is similar to the companion matrix of its characteristic polynomial.)

Let $V$ be a vector space over a scalar field $K$.

Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ be the set of vectors in $V$, where $n \geq 2$.

Then prove that the set $S$ is linearly dependent if and only if at least one of the vectors in $S$ can be written as a linear combination of remaining vectors in $S$.

Add to solve later Let $G$ be a finite group of order $21$ and let $K$ be a finite group of order $49$.

Suppose that $G$ does not have a normal subgroup of order $3$.

Then determine all group homomorphisms from $G$ to $K$.

Let $R$ be the ring of all continuous functions on the interval $[0, 2]$.

Let $I$ be the subset of $R$ defined by

\[I:=\{ f(x) \in R \mid f(1)=0\}.\]

Then prove that $I$ is an ideal of the ring $R$.

Moreover, show that $I$ is maximal and determine $R/I$.

Let $a, b$ be relatively prime integers and let $p$ be a prime number.

Suppose that we have

\[a^{2^n}+b^{2^n}\equiv 0 \pmod{p}\]
for some positive integer $n$.

Then prove that $2^{n+1}$ divides $p-1$.

Add to solve later Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$.

Let $\Aut(N)$ be the group of automorphisms of $G$.

Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime.

Then prove that $N$ is contained in the center of $G$.

Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism.

Prove that we have an isomorphism of groups:

\[G \cong \ker(f)\times \Z.\]

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

Suppose that $H < K$ and the quotient group $G/H$ is abelian.

Then prove that $G/K$ is also an abelian group.

Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$.

Then prove that the quotient group $G/N$ is also an abelian group.

Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a basis of the vector space $\R^2$, where

\[\mathbf{v}_1=\begin{bmatrix}

1 \\

1

\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}

1 \\

-1

\end{bmatrix}.\]
The action of a linear transformation $T:\R^2\to \R^3$ on the basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ is given by

\begin{align*}

T(\mathbf{v}_1)=\begin{bmatrix}

2 \\

4 \\

6

\end{bmatrix} \text{ and } T(\mathbf{v}_2)=\begin{bmatrix}

0 \\

8 \\

10

\end{bmatrix}.

\end{align*}

Find the formula of $T(\mathbf{x})$, where

\[\mathbf{x}=\begin{bmatrix}

x \\

y

\end{bmatrix}\in \R^2.\]

Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.

**(1)** \[S_1=\left \{\, \begin{bmatrix}

x_1 \\

x_2 \\

x_3

\end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\]
in the vector space $\R^3$.

x_1 \\

x_2 \\

x_3

\end{bmatrix} \in \R^3 \quad \middle | \quad x_1-4x_2+5x_3=2 \,\right \}\] in the vector space $\R^3$.

x \\

y

\end{bmatrix}\in \R^2 \quad \middle | \quad y=x^2 \quad \,\right \}\] in the vector space $\R^2$.

\[S_4=\{ f(x)\in P_4 \mid f(1) \text{ is an integer}\}\] in the vector space $P_4$.

\[S_6=\{ A\in M_{2\times 2} \mid \det(A) \neq 0\} \] in the vector space $M_{2\times 2}$.

(*Linear Algebra Exam Problem, the Ohio State University*)

\[S_8=\{ f(x)\in C[-2,2] \mid f(-1)f(1)=0\} \] in the vector space $C[-2, 2]$.

\[S_{10}=\{ f(x) \in C^2[-1, 1] \mid f^{\prime\prime}(x)+f(x)=\sin(x) \text{ for all } -1\leq x \leq 1\}\] in the vector space $C[-1, 1]$.

\[ V \setminus W = \{ \mathbf{v} \in V \mid \mathbf{v} \not\in W \}.\] Add to solve later

Let $A, B$ be complex $2\times 2$ matrices satisfying the relation

\[A=AB-BA.\]

Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.

Add to solve laterA complex square ($n\times n$) matrix $A$ is called **normal** if

\[A^* A=A A^*,\]
where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$.

A matrix $A$ is said to be **nilpotent** if there exists a positive integer $k$ such that $A^k$ is the zero matrix.

**(a)** Prove that if $A$ is both normal and nilpotent, then $A$ is the zero matrix.

You may use the fact that every normal matrix is diagonalizable.

**(b)** Give a proof of (a) without referring to eigenvalues and diagonalization.

**(c)** Let $A, B$ be $n\times n$ complex matrices. Prove that if $A$ is normal and $B$ is nilpotent such that $A+B=I$, then $A=I$, where $I$ is the $n\times n$ identity matrix.