## If the Nullity of a Linear Transformation is Zero, then Linearly Independent Vectors are Mapped to Linearly Independent Vectors

## Problem 722

Let $T: \R^n \to \R^m$ be a linear transformation.

Suppose that the nullity of $T$ is zero.

If $\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a linearly independent subset of $\R^n$, then show that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$.

Add to solve later## Find All Values of $x$ such that the Matrix is Invertible

## Problem 721

Given any constants $a,b,c$ where $a\neq 0$, find all values of $x$ such that the matrix $A$ is invertible if

\[

A=

\begin{bmatrix}

1 & 0 & c \\

0 & a & -b \\

-1/a & x & x^{2}

\end{bmatrix}

.

\]

## Find All Eigenvalues and Corresponding Eigenvectors for the $3\times 3$ matrix

## Problem 720

Find all eigenvalues and corresponding eigenvectors for the matrix $A$ if

\[

A=

\begin{bmatrix}

2 & -3 & 0 \\

2 & -5 & 0 \\

0 & 0 & 3

\end{bmatrix}

.

\]

## Find All Values of $a$ which Will Guarantee that $A$ Has Eigenvalues 0, 3, and -3.

## Problem 719

Let $A$ be the matrix given by

\[

A=

\begin{bmatrix}

-2 & 0 & 1 \\

-5 & 3 & a \\

4 & -2 & -1

\end{bmatrix}

\]
for some variable $a$. Find all values of $a$ which will guarantee that $A$ has eigenvalues $0$, $3$, and $-3$.

## Compute the Determinant of a Magic Square

## Problem 718

Let

\[

A=

\begin{bmatrix}

8 & 1 & 6 \\

3 & 5 & 7 \\

4 & 9 & 2

\end{bmatrix}

.

\]
Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic square.

Compute the determinant of $A$.

Add to solve later## Are These Linear Transformations?

## Problem 717

Define two functions $T:\R^{2}\to\R^{2}$ and $S:\R^{2}\to\R^{2}$ by

\[

T\left(

\begin{bmatrix}

x \\ y

\end{bmatrix}

\right)

=

\begin{bmatrix}

2x+y \\ 0

\end{bmatrix}

,\;

S\left(

\begin{bmatrix}

x \\ y

\end{bmatrix}

\right)

=

\begin{bmatrix}

x+y \\ xy

\end{bmatrix}

.

\]
Determine whether $T$, $S$, and the composite $S\circ T$ are linear transformations.

## Using Gram-Schmidt Orthogonalization, Find an Orthogonal Basis for the Span

## Problem 716

Using Gram-Schmidt orthogonalization, find an orthogonal basis for the span of the vectors $\mathbf{w}_{1},\mathbf{w}_{2}\in\R^{3}$ if

\[

\mathbf{w}_{1}

=

\begin{bmatrix}

1 \\ 0 \\ 3

\end{bmatrix}

,\quad

\mathbf{w}_{2}

=

\begin{bmatrix}

2 \\ -1 \\ 0

\end{bmatrix}

.

\]

## Normalize Lengths to Obtain an Orthonormal Basis

## Problem 715

Let

\[

\mathbf{v}_{1}

=

\begin{bmatrix}

1 \\ 1

\end{bmatrix}

,\;

\mathbf{v}_{2}

=

\begin{bmatrix}

1 \\ -1

\end{bmatrix}

.

\]
Let $V=\Span(\mathbf{v}_{1},\mathbf{v}_{2})$. Do $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ form an orthonormal basis for $V$?

If not, then find an orthonormal basis for $V$.

Add to solve later## Find a Spanning Set for the Vector Space of Skew-Symmetric Matrices

## Problem 714

Let $W$ be the set of $3\times 3$ skew-symmetric matrices. Show that $W$ is a subspace of the vector space $V$ of all $3\times 3$ matrices. Then, exhibit a spanning set for $W$.

Add to solve later## Determine Bases for Nullspaces $\calN(A)$ and $\calN(A^{T}A)$

## Problem 713

Determine bases for $\calN(A)$ and $\calN(A^{T}A)$ when

\[

A=

\begin{bmatrix}

1 & 2 & 1 \\

1 & 1 & 3 \\

0 & 0 & 0

\end{bmatrix}

.

\]
Then, determine the ranks and nullities of the matrices $A$ and $A^{\trans}A$.

## In which $\R^k$, are the Nullspace and Range Subspaces?

## Problem 712

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

Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$.

## Prove Vector Space Properties Using Vector Space Axioms

## Problem 711

Using the axiom of a vector space, prove the following properties.

Let $V$ be a vector space over $\R$. Let $u, v, w\in V$.

**(a)** If $u+v=u+w$, then $v=w$.

**(b)** If $v+u=w+u$, then $v=w$.

**(c)** The zero vector $\mathbf{0}$ is unique.

**(d)** For each $v\in V$, the additive inverse $-v$ is unique.

**(e)** $0v=\mathbf{0}$ for every $v\in V$, where $0\in\R$ is the zero scalar.

**(f)** $a\mathbf{0}=\mathbf{0}$ for every scalar $a$.

**(g)** If $av=\mathbf{0}$, then $a=0$ or $v=\mathbf{0}$.

**(h)** $(-1)v=-v$.

The first two properties are called the **cancellation law**.

## Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors

## Problem 710

Find a basis for $\Span(S)$ where $S=

\left\{

\begin{bmatrix}

1 \\ 2 \\ 1

\end{bmatrix}

,

\begin{bmatrix}

-1 \\ -2 \\ -1

\end{bmatrix}

,

\begin{bmatrix}

2 \\ 6 \\ -2

\end{bmatrix}

,

\begin{bmatrix}

1 \\ 1 \\ 3

\end{bmatrix}

\right\}$.

## Find a Basis for the Subspace spanned by Five Vectors

## Problem 709

Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where

\[

\mathbf{v}_{1}=

\begin{bmatrix}

1 \\ 2 \\ 2 \\ -1

\end{bmatrix}

,\;\mathbf{v}_{2}=

\begin{bmatrix}

1 \\ 3 \\ 1 \\ 1

\end{bmatrix}

,\;\mathbf{v}_{3}=

\begin{bmatrix}

1 \\ 5 \\ -1 \\ 5

\end{bmatrix}

,\;\mathbf{v}_{4}=

\begin{bmatrix}

1 \\ 1 \\ 4 \\ -1

\end{bmatrix}

,\;\mathbf{v}_{5}=

\begin{bmatrix}

2 \\ 7 \\ 0 \\ 2

\end{bmatrix}

.\]
Find a basis for the span $\Span(S)$.

## How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix

## Problem 708

Let $A=\begin{bmatrix}

2 & 4 & 6 & 8 \\

1 &3 & 0 & 5 \\

1 & 1 & 6 & 3

\end{bmatrix}$.

**(a)** Find a basis for the nullspace of $A$.

**(b)** Find a basis for the row space of $A$.

**(c)** Find a basis for the range of $A$ that consists of column vectors of $A$.

**(d)** For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.

## Can We Reduce the Number of Vectors in a Spanning Set?

## Problem 707

Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^3$. Is it possible that $S_2=\{\mathbf{v}_1\}$ is a spanning set for $V$?

Add to solve later## Does an Extra Vector Change the Span?

## Problem 706

Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set

\[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\]
still a spanning set for $V$? If so, prove it. Otherwise, give a counterexample.

## Vector Space of Functions from a Set to a Vector Space

## Problem 705

For a set $S$ and a vector space $V$ over a scalar field $\K$, define the set of all functions from $S$ to $V$

\[ \Fun ( S , V ) = \{ f : S \rightarrow V \} . \]

For $f, g \in \Fun(S, V)$, $z \in \K$, addition and scalar multiplication can be defined by

\[ (f+g)(s) = f(s) + g(s) \, \mbox{ and } (cf)(s) = c (f(s)) \, \mbox{ for all } s \in S . \]

**(a)** Prove that $\Fun(S, V)$ is a vector space over $\K$. What is the zero element?

**(b)** Let $S_1 = \{ s \}$ be a set consisting of one element. Find an isomorphism between $\Fun(S_1 , V)$ and $V$ itself. Prove that the map you find is actually a linear isomorpism.

**(c)** Suppose that $B = \{ e_1 , e_2 , \cdots , e_n \}$ is a basis of $V$. Use $B$ to construct a basis of $\Fun(S_1 , V)$.

**(d)** Let $S = \{ s_1 , s_2 , \cdots , s_m \}$. Construct a linear isomorphism between $\Fun(S, V)$ and the vector space of $n$-tuples of $V$, defined as

\[ V^m = \{ (v_1 , v_2 , \cdots , v_m ) \mid v_i \in V \mbox{ for all } 1 \leq i \leq m \} . \]

**(e)** Use the basis $B$ of $V$ to constract a basis of $\Fun(S, V)$ for an arbitrary finite set $S$. What is the dimension of $\Fun(S, V)$?

**(f)** Let $W \subseteq V$ be a subspace. Prove that $\Fun(S, W)$ is a subspace of $\Fun(S, V)$.

## Find a Basis for Nullspace, Row Space, and Range of a Matrix

## Problem 704

Let $A=\begin{bmatrix}

2 & 4 & 6 & 8 \\

1 &3 & 0 & 5 \\

1 & 1 & 6 & 3

\end{bmatrix}$.

**(a)** Find a basis for the nullspace of $A$.

**(b)** Find a basis for the row space of $A$.

**(c)** Find a basis for the range of $A$ that consists of column vectors of $A$.

**(d)** For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.

## Describe the Range of the Matrix Using the Definition of the Range

## Problem 703

Using the definition of the range of a matrix, describe the range of the matrix

\[A=\begin{bmatrix}

2 & 4 & 1 & -5 \\

1 &2 & 1 & -2 \\

1 & 2 & 0 & -3

\end{bmatrix}.\]