## Prove that any Set of Vectors Containing the Zero Vector is Linearly Dependent

## Problem 652

Prove that any set of vectors which contains the zero vector is linearly dependent.

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Prove that any set of vectors which contains the zero vector is linearly dependent.

Add to solve later Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$.

Consider the functions \[f(x)=\sin^2(x) \text{ and } g(x)=\cos^2(x)\]
in $C[-2\pi, 2\pi]$.

Prove or disprove that the functions $f(x)$ and $g(x)$ are linearly independent.

*(The Ohio State University, Linear Algebra Midterm)*

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**(a)** Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent.

**(b)** Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.

Let $V$ be a subspace of $\R^n$.

Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a basis of the subspace $V$.

Prove that every basis of $V$ consists of $k$ vectors in $V$.

Add to solve later Let $V$ be a subspace of $\R^n$.

Suppose that

\[S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}\]
is a spanning set for $V$.

Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.

Add to solve laterThe following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.

There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).

The time limit was 55 minutes.

This post is Part 3 and contains Problem 7, 8, and 9.

Check out Part 1 and Part 2 for the rest of the exam problems.

**Problem 7**. Let $A=\begin{bmatrix}

-3 & -4\\

8& 9

\end{bmatrix}$ and $\mathbf{v}=\begin{bmatrix}

-1 \\

2

\end{bmatrix}$.

**(a)** Calculate $A\mathbf{v}$ and find the number $\lambda$ such that $A\mathbf{v}=\lambda \mathbf{v}$.

**(b)** Without forming $A^3$, calculate the vector $A^3\mathbf{v}$.

**Problem 8**. Prove that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.

**Problem 9**.

Determine whether each of the following sentences is true or false.

**(a)** There is a $3\times 3$ homogeneous system that has exactly three solutions.

**(b)** If $A$ and $B$ are $n\times n$ symmetric matrices, then the sum $A+B$ is also symmetric.

**(c)** If $n$-dimensional vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are linearly dependent, then the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$ is also linearly dependent for any $n$-dimensional vector $\mathbf{v}_4$.

**(d)** If the coefficient matrix of a system of linear equations is singular, then the system is inconsistent.

**(e)** The vectors

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

1 \\

0 \\

1

\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}

0 \\

0 \\

1

\end{bmatrix}\]
are linearly independent.

Let

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

1 \\

2 \\

0

\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}

1 \\

a \\

5

\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}

0 \\

4 \\

b

\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.

Add to solve later 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 $\mathbf{b}$.

Add to solve later**(a)** Let $R$ be a commutative ring. If we regard $R$ as a left $R$-module, then prove that any two distinct elements of the module $R$ are linearly dependent.

**(b)** Let $f: M\to M’$ be a left $R$-module homomorphism. Let $\{x_1, \dots, x_n\}$ be a subset in $M$. Prove that if the set $\{f(x_1), \dots, f(x_n)\}$ is linearly independent, then the set $\{x_1, \dots, x_n\}$ is also linearly independent.

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Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$.

Let $A \in V$ and consider the set

\[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\]
of $n^2$ elements.

Prove that the set $S_A$ cannot be a basis of the vector space $V$ for any $A\in V$.

Let $f(x)=\sin^2(x)$, $g(x)=\cos^2(x)$, and $h(x)=1$. These are vectors in $C[-1, 1]$.

Determine whether the set $\{f(x), \, g(x), \, h(x)\}$ is linearly dependent or linearly independent.

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

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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 $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$.

Then prove that $V$ is a subspace of $\R^n$.

Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be $2$-dimensional vectors and let $A$ be a $2\times 2$ matrix.

**(a)** Show that if $\mathbf{v}_1, \mathbf{v}_2$ are linearly dependent vectors, then the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly dependent.

**(b)** If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors, can we conclude that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent?

**(c)** If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors and $A$ is nonsingular, then show that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent.

**(a)** For what value(s) of $a$ is the following set $S$ linearly dependent?

\[ S=\left \{\,\begin{bmatrix}

1 \\

2 \\

3 \\

a

\end{bmatrix}, \begin{bmatrix}

a \\

0 \\

-1 \\

2

\end{bmatrix}, \begin{bmatrix}

0 \\

0 \\

a^2 \\

7

\end{bmatrix}, \begin{bmatrix}

1 \\

a \\

1 \\

1

\end{bmatrix}, \begin{bmatrix}

2 \\

-2 \\

3 \\

a^3

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

**(b)** Let $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of nonzero vectors in $\R^m$ such that the dot product

\[\mathbf{v}_i\cdot \mathbf{v}_j=0\]
when $i\neq j$.

Prove that the set is linearly independent.

Determine whether the following set of vectors is linearly independent or linearly dependent. If the set is linearly dependent, express one vector in the set as a linear combination of the others.

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

1 \\

0 \\

-1 \\

0

\end{bmatrix}, \begin{bmatrix}

1 \\

2 \\

3 \\

4

\end{bmatrix}, \begin{bmatrix}

-1 \\

-2 \\

0 \\

1

\end{bmatrix},

\begin{bmatrix}

-2 \\

-2 \\

7 \\

11

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

Let $V$ be an $n$-dimensional vector space over a field $K$.

Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ are linearly independent vectors in $V$.

Are the following vectors linearly independent?

\[\mathbf{v}_1+\mathbf{v}_2, \quad \mathbf{v}_2+\mathbf{v}_3, \quad \dots, \quad \mathbf{v}_{k-1}+\mathbf{v}_k, \quad \mathbf{v}_k+\mathbf{v}_1.\]

If it is linearly dependent, give a non-trivial linear combination of these vectors summing up to the zero vector.

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