Linear Algebra Examples

$[\begin{array}{c} 0 + 0 i \\ 2 + 1 i \\ 2 + 1 i \end{array}]$

Step 1

The norm is the square root of the sum of squares of each element in the vector.

$\sqrt{{| 0 + 0 i |}^{2} + {| 2 + 1 i |}^{2} + {| 2 + 1 i |}^{2}}$

Step 2

Simplify.

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Step 2.1

Multiply $0$ by $i$ .

$\sqrt{{| 0 + 0 |}^{2} + {| 2 + 1 i |}^{2} + {| 2 + 1 i |}^{2}}$

Step 2.2

Add $0$ and $0$ .

$\sqrt{{| 0 |}^{2} + {| 2 + 1 i |}^{2} + {| 2 + 1 i |}^{2}}$

Step 2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$\sqrt{0^{2} + {| 2 + 1 i |}^{2} + {| 2 + 1 i |}^{2}}$

Step 2.4

Raising $0$ to any positive power yields $0$ .

$\sqrt{0 + {| 2 + 1 i |}^{2} + {| 2 + 1 i |}^{2}}$

Step 2.5

Multiply $i$ by $1$ .

$\sqrt{0 + {| 2 + i |}^{2} + {| 2 + 1 i |}^{2}}$

Step 2.6

Use the formula $| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

$\sqrt{0 + {\sqrt{2^{2} + 1^{2}}}^{2} + {| 2 + 1 i |}^{2}}$

Step 2.7

Raise $2$ to the power of $2$ .

$\sqrt{0 + {\sqrt{4 + 1^{2}}}^{2} + {| 2 + 1 i |}^{2}}$

Step 2.8

One to any power is one.

$\sqrt{0 + {\sqrt{4 + 1}}^{2} + {| 2 + 1 i |}^{2}}$

Step 2.9

Add $4$ and $1$ .

$\sqrt{0 + {\sqrt{5}}^{2} + {| 2 + 1 i |}^{2}}$

Step 2.10

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{0 + {(5^{\frac{1}{2}})}^{2} + {| 2 + 1 i |}^{2}}$

Step 2.10.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{0 + 5^{\frac{1}{2} \cdot 2} + {| 2 + 1 i |}^{2}}$

Step 2.10.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{0 + 5^{\frac{2}{2}} + {| 2 + 1 i |}^{2}}$

Step 2.10.4

Cancel the common factor of $2$ .

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Step 2.10.4.1

Cancel the common factor.

$\sqrt{0 + 5^{\frac{2}{2}} + {| 2 + 1 i |}^{2}}$

Step 2.10.4.2

Rewrite the expression.

$\sqrt{0 + 5^{1} + {| 2 + 1 i |}^{2}}$

Step 2.10.5

Evaluate the exponent.

$\sqrt{0 + 5 + {| 2 + 1 i |}^{2}}$

Step 2.11

Multiply $i$ by $1$ .

$\sqrt{0 + 5 + {| 2 + i |}^{2}}$

Step 2.12

Use the formula $| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

$\sqrt{0 + 5 + {\sqrt{2^{2} + 1^{2}}}^{2}}$

Step 2.13

Raise $2$ to the power of $2$ .

$\sqrt{0 + 5 + {\sqrt{4 + 1^{2}}}^{2}}$

Step 2.14

One to any power is one.

$\sqrt{0 + 5 + {\sqrt{4 + 1}}^{2}}$

Step 2.15

Add $4$ and $1$ .

$\sqrt{0 + 5 + {\sqrt{5}}^{2}}$

Step 2.16

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.16.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{0 + 5 + {(5^{\frac{1}{2}})}^{2}}$

Step 2.16.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{0 + 5 + 5^{\frac{1}{2} \cdot 2}}$

Step 2.16.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{0 + 5 + 5^{\frac{2}{2}}}$

Step 2.16.4

Cancel the common factor of $2$ .

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Step 2.16.4.1

Cancel the common factor.

$\sqrt{0 + 5 + 5^{\frac{2}{2}}}$

Step 2.16.4.2

Rewrite the expression.

$\sqrt{0 + 5 + 5^{1}}$

Step 2.16.5

Evaluate the exponent.

$\sqrt{0 + 5 + 5}$

Step 2.17

Add $0$ and $5$ .

$\sqrt{5 + 5}$

Step 2.18

Add $5$ and $5$ .

$\sqrt{10}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{10}$

Decimal Form:

$3.16227766 \dots$