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Linear Algebra : Norms

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Example Question #11 : Norms

Find the norm of the vector $\vec{v} = (3, -2, -2)$

Possible Answers:

$\sqrt{15}$

$\sqrt{17}$

$\sqrt{7}$

Correct answer:

$\sqrt{17}$

Explanation:

$|| \vec{v} || = \sqrt{3^2 + (-2)^2 + (-2)^2 } = \sqrt{9+4+4} = \sqrt{17}$

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Example Question #12 : Norms

Find the norm of the vector $\vec{v} = (4, -1, 3 )$

Possible Answers:

$\sqrt{26}$

$\sqrt{24}$

$\sqrt{6}$

Correct answer:

$\sqrt{26}$

Explanation:

$|| \vec{v} || = \sqrt{ 4^2 + (-1)^2 + 3^2 } = \sqrt{16+1+9 } = \sqrt{26}$

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Example Question #13 : Norms

Find the norm of the vector $\vec{v} = (3, 4, 5 )$

Possible Answers:

$5 \sqrt{2}$

$2\sqrt{5}$

$2 \sqrt{3}$

$3 \sqrt{2}$

Correct answer:

$5 \sqrt{2}$

Explanation:

$|| \vec{v} || = \sqrt{ 3^2 + 4^2 + 5^2 } = \sqrt{9+16+25} = \sqrt{50 }$

This can be simplified:

$\sqrt{2 \cdot 5 \cdot 5 } = 5 \sqrt{2 }$

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Example Question #14 : Norms

Let $\textbf{u} = (2, 3, a, 1 )$ for some real number .

Give such that $\left \| \textbf{u} \right \| = 4$ .

Possible Answers:

$\pm \frac{16}{13}$

$\frac{2}{3}$

$\pm \sqrt{2}$

$\pm \frac{4 \sqrt{13}}{13}$

Correct answer:

$\pm \sqrt{2}$

Explanation:

$\left \| \textbf{u} \right \|$ , the norm, or length, of vector $\textbf{u}$ , is equal to the square root of the sum of the squares of its elements. Therefore,

$\left \| \textbf{u} \right \| = \sqrt{2^{2}+3^{2}+a^{2}+1^{2}}$

$= \sqrt{4+9+a^{2}+1}$

$= \sqrt{ a^{2}+14}$

Set this equal to 4:

$\sqrt{ a^{2}+14} = 4$

$a^{2}+14 = 16$

$a^{2}= 2$

$a = \pm \sqrt{2}$

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Example Question #15 : Norms

$\textbf{u}= (-3, 2, x, x)$ ,

where is a real number.

In terms of , give $\left \| \textbf{u} \right \|$ .

Possible Answers:

$\sqrt{2 x ^{2} +13}$

$x \sqrt{6}$

$\sqrt{2 x ^{2} -5}$

$\sqrt{2x+5}$

$\sqrt{2x-1}$

Correct answer:

$\sqrt{2 x ^{2} +13}$

Explanation:

$\left \| \textbf{u} \right \|$ , the norm, or length, of vector $\textbf{u}$ , is equal to the square root of the sum of the squares of its elements. Therefore,

$\left \| \textbf{u} \right \|= \sqrt{(-3)^{2}+ 2 ^{2}+ x ^{2} +x ^{2}}$

$=\sqrt{9+4+ x ^{2} +x ^{2}}$

$=\sqrt{2 x ^{2} +13}$

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Example Question #12 : Norms

$\bold{v} = \left ( \frac{1}{3}, - \frac{1}{3},\frac{1}{3} \right )$

True or false: $\textbf{v}$ is an example of a unit vector.

Possible Answers:

True

False

Correct answer:

False

Explanation:

$\textbf{v}$ is a unit vector if and only if its norm, or length, $\left \| \textbf{v} \right \|$ - the square root of the sum of the squares of its elements - is equal to 1. Find the length using this definition:

$\left \| \textbf{v} \right \| = \sqrt{\left ( \frac{1}{3} \right )^{2}+ \left ( -\frac{1}{3} \right )^{2}+ \left ( \frac{1}{3} \right )^{2}}$

$= \sqrt{\frac{1}{9}+ \frac{1}{9} + \frac{1}{9} }$

$= \sqrt{\frac{1}{3} }$

$\left \| \textbf{v} \right \| \ne 1$ , so $\textbf{v}$ is not a unit vector.

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Example Question #17 : Norms

$\bold{v} = \left ( \frac{1}{2},\frac{ \sqrt{2}}{2}, -\frac{1}{2} \right )$

True or false: $\textbf{v}$ is an example of a unit vector.

Possible Answers:

False

True

Correct answer:

True

Explanation:

$\left \| \textbf{v} \right \| = \sqrt{\left ( \frac{1}{2} \right )^{2}+ \left ( \frac{\sqrt{2}}{2} \right ) ^{2}+ \left ( -\frac{1}{2} \right ) ^{2}}$

$=\sqrt{ \frac{1}{4} + \frac{2}{4} + \frac{1}{4} }$

$=\sqrt{1}$

$\textbf{v}$ is a unit vector.

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Example Question #18 : Norms

Which of these functions could be that of a Euclidean norm operator? You may assume each function is onto.

Possible Answers:

All of the other answers are norm operators

$h: \mathbb{N}^3 \rightarrow \mathbb{Q}$

$f:\mathbb{R}^7 \rightarrow \mathbb{R}$

$g: \mathbb{Q}^2 \rightarrow \mathbb{N}$

Correct answer:

$f:\mathbb{R}^7 \rightarrow \mathbb{R}$

Explanation:

This function's range is $\mathbb{R}$ , the set of all real numbers. In short, this is set of all possible "distances between two given numbers" in elementary linear algebra. would not be a norm. For example, $h(1,1,1) = \sqrt{1+1+1} =\sqrt{3}$ , which is not a rational number (part of $\mathbb{Q}$ ). Similarly, is also not a norm. We have $g(1,1) = \sqrt{1+1} = \sqrt{2}$ , which is not a natural number.

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Example Question #19 : Norms

The taxicab norm on $\mathbb{R}^n$ for a vector (x_1,...,x_n)^T is defined as

||x||_1=|x_1|+...+|x_n|

Given v=(1,2,-3)^T , find ||v||_1 .

Possible Answers:

||v||_1=1

||v||_1=6

||v||_1=0

$||v||_1=\sqrt{6}$

Correct answer:

||v||_1=6

Explanation:

To find ||v||_1 given v=(1,2,-3)^T , we simply do what the taxicab norm formula tells us:

||v||_1=|1|+|2|+|-3|=1+2+3=6

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Example Question #14 : Norms

Find the euclidean norm of the vector $(\pi,e)$

Possible Answers:

$\sqrt{\pi^2+e^2}$

$\pi^2+e^2$

$\sqrt{\pi+e}$

$\pi+e$

Correct answer:

$\sqrt{\pi^2+e^2}$

Explanation:

To find the euclidean norm of $(\pi,e)$ , we take the sum of the entries squared and take the square root:

$\sqrt{\pi^2+e^2}$

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Linear Algebra : Norms

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #11 : Norms

Example Question #12 : Norms

Example Question #13 : Norms

Example Question #14 : Norms

Example Question #15 : Norms

Example Question #12 : Norms

Example Question #17 : Norms

Example Question #18 : Norms

Example Question #19 : Norms

Example Question #14 : Norms

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