## A Basis for the Vector Space of Polynomials of Degree Two or Less and Coordinate Vectors

## Problem 150

Show that the set

\[S=\{1, 1-x, 3+4x+x^2\}\]
is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less.

Show that the set

\[S=\{1, 1-x, 3+4x+x^2\}\]
is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less.

Let $A$ be an $n \times n$ nilpotent matrix, that is, $A^m=O$ for some positive integer $m$, where $O$ is the $n \times n$ zero matrix.

Prove that $A$ is a singular matrix and also prove that $I-A, I+A$ are both nonsingular matrices, where $I$ is the $n\times n$ identity matrix.

Add to solve laterLet $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.

**(a) **The set $S$ consisting of all $n\times n$ symmetric matrices.

**(b)** The set $T$ consisting of all $n \times n$ skew-symmetric matrices.

**(c)** The set $U$ consisting of all $n\times n$ nonsingular matrices.

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)$.

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 $\mathbf{u}_{n+1} \not \in U$.

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

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 $i=1,2, \dots, m$.

Find the determinant of the matix

\[A=\begin{bmatrix}

100 & 101 & 102 \\

101 &102 &103 \\

102 & 103 & 104

\end{bmatrix}.\]

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 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 $A$ 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 $A$ 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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