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Linear Algebra : Operations and Properties

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Example Questions

Example Question #207 : Linear Algebra

$B= \begin{bmatrix} 1& -2 & 3 &- 4 & 5 \\ -2 & 3 &-4 & 5 & -1 \\ 3 & -4 & 5 & -1 & 2 \\-4 & 5 &- 1 & 2 & -3 \\5 &-1 & 2 & -3& 4 \end{bmatrix}$

Give the trace of .

Possible Answers:

120

Correct answer:

Explanation:

The trace of a matrix is the sum of the elements along its main diagonal, which are shown below:

$B= \begin{bmatrix} \textbf{{\color{Red} 1}}&- 2 & 3 & -4 & 5 \\ -2 &\textbf{{\color{Red} 3}} & -4 & 5 & -1 \\ 3 & -4 & \textbf{{\color{Red} 5}} & -1 & 2 \\-4 & 5 &- 1 & \textbf{{\color{Red} 2}} & -3 \\5 & -1 & 2 &- 3& \textbf{{\color{Red} 4}} \end{bmatrix}$

$\textup{Tr}(B)= 1+ 3+ 5+ 2 + 4 = 15$ .

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Example Question #208 : Linear Algebra

$B= \begin{bmatrix} 1& 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 1 \\ 3 & 4 & 5 & 1 & 2 \\4 & 5 & 1 & 2 & 3 \\5 & 1 & 2 & 3& 4 \end{bmatrix}$

Give the trace of .

Possible Answers:

120

Correct answer:

Explanation:

The trace of a matrix is the sum of the elements along its main diagonal, which are shown below:

$B= \begin{bmatrix} \textbf{{\color{Red} 1}}& 2 & 3 & 4 & 5 \\ 2 &\textbf{{\color{Red} 3}} & 4 & 5 & 1 \\ 3 & 4 & \textbf{{\color{Red} 5}} & 1 & 2 \\4 & 5 & 1 & \textbf{{\color{Red} 2}} & 3 \\5 & 1 & 2 & 3& \textbf{{\color{Red} 4}} \end{bmatrix}$

$\textup{Tr}(B)= 1+ 3+ 5+ 2 + 4 = 15$ .

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Example Question #131 : Operations And Properties

Find the trace of the matrix $\begin{bmatrix} 0& 1& 1& 1\\ 1& 0&1 &1 \\ 1&1 & 1& 1\\ 1&1 & 1& 1 \end{bmatrix}$ .

Possible Answers:

None of the other answers

Correct answer:

Explanation:

To find the trace of a matrix, we simply add up the entries along the main diagonal (Start with the top-left and work your way down to the bottom right). So the trace of this matrix is 1+1+0+0=2 .

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Example Question #132 : Operations And Properties

Which of the following $n\times n$ matrices is guaranteed NOT to have a trace of ?

Possible Answers:

Any skew-symmetric matrix

A linear transformation (mapping) matrix

The identity matrix

The zero matrix

Any symmetric matrix

Correct answer:

The identity matrix

Explanation:

The identity matrix of any size has along its main diagonal, and thus cannot have a trace of .

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Example Question #1 : Norms

Find the norm of the following vector.

A=[3,4,5]

Possible Answers:

$\left \| A\right \|=\sqrt{52}$

$\left \| A\right \|=5$

$\left \| A\right \|=2\sqrt{5}$

$\left \| A\right \|=\sqrt{2}$

$\left \| A\right \|=5\sqrt{2}$

Correct answer:

$\left \| A\right \|=5\sqrt{2}$

Explanation:

The norm of a vector is simply the square root of the sum of each component squared.

$\left \| A\right \|=\sqrt{3^2+4^2+5^2}=\sqrt{9+16+25}=\sqrt{50}=5\sqrt{2}$

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Example Question #1 : Norms

Find the norm of vector .

$A=(\cos(x), \sin(x))$

Possible Answers:

$\left \| A\right \|=1$

$\left \| A\right \|=\cos(x)+\sin(x)$

$\left \| A\right \|=0$

$\left \| A\right \|=1+\cos^2(x)$

$\left \| A\right \|=1+\sin^2(x)$

Correct answer:

$\left \| A\right \|=1$

Explanation:

In order to find the norm, we need to square each component, sum them up, and then take the square root.

$\left \| A\right \|=\sqrt{\cos^2(x)+\sin^2(x)}=\sqrt{1}=1$

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Example Question #1 : Norms

Find the norm, $\begin{Vmatrix} \bf{a} \end{Vmatrix}$ , given $\bf{a}= \begin{bmatrix} 2\\ 1\\ 7\\ -2 \end{bmatrix}$

Possible Answers:

$\begin{Vmatrix} \bf{a} \end{Vmatrix}=\sqrt{8}$

$\begin{Vmatrix} \bf{a} \end{Vmatrix}=\sqrt{58}$

$\begin{Vmatrix} \bf{a} \end{Vmatrix}=58$

$\begin{Vmatrix} \bf{a} \end{Vmatrix}=-28$

Correct answer:

$\begin{Vmatrix} \bf{a} \end{Vmatrix}=\sqrt{58}$

Explanation:

By definition,

$\begin{Vmatrix} \bf{a} \end{Vmatrix}= \sqrt{a_1^2+a_2^2+a_3^2+...+a_n^2}$ ,

therefore,

$\begin{Vmatrix} \bf{a} \end{Vmatrix}= \sqrt{2^2+1^2+7^2+(-2)^2}= \sqrt{58}$ .

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Example Question #1 : Norms

Calculate the norm of $\bf{A}\times \bf{B}$ , or $\begin{Vmatrix} \bf{A}\times \bf{B} \end{Vmatrix}$ , given

$\bf{A} = 3 \hat{i} - 1 \hat{j}$ ,

$\bf{B} = -1 \hat{k}$ .

Possible Answers:

Can not be determined.

$\begin{Vmatrix} \bf{A}\times \bf{B} \end{Vmatrix}= 10$

$\begin{Vmatrix} \bf{A}\times \bf{B} \end{Vmatrix}= 4$

$\begin{Vmatrix} \bf{A}\times \bf{B} \end{Vmatrix}=\sqrt{10}$

Correct answer:

$\begin{Vmatrix} \bf{A}\times \bf{B} \end{Vmatrix}=\sqrt{10}$

Explanation:

First, we need to find $\bf{A}\times \bf{B}$ . This is, by definition,

$\bf{A \times \bf{B}}= \begin{vmatrix} \hat{i}&\hat{j} &\hat{k} \\ 3& -1& 0\\ 0&0 &-1 \end{vmatrix}= 1\hat{i}+3\hat{j}$ .

Therefore,

$\begin{Vmatrix} \bf{A}\times \bf{B} \end{Vmatrix}=\sqrt{1^2+3^2}=\sqrt{10}$ .

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Example Question #2 : Norms

Find the norm of the vector $\vec{v} = (4, 3, 0)$

Possible Answers:

$\sqrt{7}$

Correct answer:

Explanation:

To find the norm, square each component, add, then take the square root:

$\sqrt{4^2 + 3^2 + 0^2 } = \sqrt{16 + 9 } = \sqrt{25 } = 5$

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Example Question #1 : Norms

Find a unit vector in the same direction as $\vec{v} = (2, -3, \sqrt{3})$

Possible Answers:

(4,4,4)

$(\frac{2}{\sqrt{2}}, -\frac{3}{\sqrt{2}}, \sqrt{\frac{3}{2}})$

$(\frac{1}{2} , -\frac{3}{4} , \frac{\sqrt{3}}{4})$

$(1, -1.5, \frac{\sqrt3}{2} )$

Correct answer:

$(\frac{1}{2} , -\frac{3}{4} , \frac{\sqrt{3}}{4})$

Explanation:

First, find the length of the vector: $\sqrt{ 2^2 + (-3)^2 + (\sqrt{3})^2 } = \sqrt{4+9+3} = \sqrt{16} = 4$

Because this vector has the length of 4 and a unit vector would have a length of 1, divide everything by 4:

$( \frac{ 2}{4} , \frac{-3}{4} , \frac{\sqrt{3}}{4}) = (\frac{1}{2} , -\frac{3}{4} , \frac{\sqrt{3}}{4})$

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Linear Algebra : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #207 : Linear Algebra

Example Question #208 : Linear Algebra

Example Question #131 : Operations And Properties

Example Question #132 : Operations And Properties

Example Question #1 : Norms

Example Question #1 : Norms

Example Question #1 : Norms

Example Question #1 : Norms

Example Question #2 : Norms

Example Question #1 : Norms

All Linear Algebra Resources

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