## 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.

Add to solve laterProve that any set of vectors which contains the zero vector is linearly dependent.

Add to solve later**(a) **Find a function

\[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\]
such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants.

**(b)** Find real numbers $a, b, c$ such that the function

\[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\]
satisfies $g(0) = 3$, $g(\pi/2) = 1$, and $g(\pi) = -5$.

Find a quadratic function $f(x) = ax^2 + bx + c$ such that $f(1) = 3$, $f'(1) = 3$, and $f^{\prime\prime}(1) = 2$.

Here, $f'(x)$ and $f^{\prime\prime}(x)$ denote the first and second derivatives, respectively.

Add to solve laterA 2-digit number has two properties: The digits sum to 11, and if the number is written with digits reversed, and subtracted from the original number, the result is 45.

Find the number.

Add to solve laterDetermine whether the following augmented matrices are in reduced row echelon form, and calculate the solution sets of their associated systems of linear equations.

**(a)** $\left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 6 \end{array} \right]$.

**(b)** $\left[\begin{array}{rrr|r} 1 & 0 & 3 & -4 \\ 0 & 1 & 2 & 0 \end{array} \right]$.

**(c)** $\left[\begin{array}{rr|r} 1 & 2 & 0 \\ 1 & 1 & -1 \end{array} \right]$.

Read solution

Recall that a matrix $A$ is **symmetric** if $A^\trans = A$, where $A^\trans$ is the transpose of $A$.

Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it.

Otherwise, provide a counterexample.

Add to solve later**(a)** Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.

**(b)** Find all such matrices with rank 2.

Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix.

Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.

Read solution

If $A, B$ have the same rank, can we conclude that they are row-equivalent?

If so, then prove it. If not, then provide a counterexample.

Add to solve laterFor each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix.

**(a) **$A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$.

**(b)** $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$.

**(c)** $C = \begin{bmatrix} 2 & -2 & 4 \\ 4 & 1 & -2 \\ 6 & -1 & 2 \end{bmatrix}$.

**(d)** $D = \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}$.

**(e)** $E = \begin{bmatrix} -2 & 3 & 1 \end{bmatrix}$.

If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$?

If so, then prove it. If not, then provide a counterexample.

Add to solve laterLet $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$.

Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$.

Add to solve laterLet $\mathbf{v}$ be an $n \times 1$ column vector.

Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix.

Add to solve laterLet $\mathbf{v}$ be an $n \times 1$ column vector.

Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$.

Add to solve laterLet $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors.

Prove that $\tr ( \mathbf{v} \mathbf{w}^\trans ) = \mathbf{v}^\trans \mathbf{w}$.

Add to solve laterLet $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors.

**(a)** Prove that $\mathbf{v}^\trans \mathbf{w} = \mathbf{w}^\trans \mathbf{v}$.

**(b)** Provide an example to show that $\mathbf{v} \mathbf{w}^\trans$ is not always equal to $\mathbf{w} \mathbf{v}^\trans$.

Calculate the following expressions, using the following matrices:

\[A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}\]

**(a)** $A B^\trans + \mathbf{v} \mathbf{v}^\trans$.

**(b)** $A \mathbf{v} – 2 \mathbf{v}$.

**(c)** $\mathbf{v}^{\trans} B$.

**(d)** $\mathbf{v}^\trans \mathbf{v} + \mathbf{v}^\trans B A^\trans \mathbf{v}$.

Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector.

Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$.

Add to solve laterLet $A$ and $B$ be $n \times n$ matrices.

Is it always true that $\tr (A B) = \tr (A) \tr (B) $?

If it is true, prove it. If not, give a counterexample.

Add to solve laterLet $A$ be an $n \times n$ matrix.

Is it true that $\tr ( A^\trans ) = \tr(A)$? If it is true, prove it. If not, give a counterexample.

Add to solve later