Linear Algebra Examples

$[\begin{array}{c} 3 - 2 i \\ 3 - 2 i \\ 0 - 1 i \end{array}]$

Step 1

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

$\sqrt{{| 3 - 2 i |}^{2} + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2

Simplify.

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Step 2.1

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

$\sqrt{{\sqrt{3^{2} + {(- 2)}^{2}}}^{2} + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.2

Raise $3$ to the power of $2$ .

$\sqrt{{\sqrt{9 + {(- 2)}^{2}}}^{2} + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.3

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

$\sqrt{{\sqrt{9 + 4}}^{2} + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.4

Add $9$ and $4$ .

$\sqrt{{\sqrt{13}}^{2} + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.5

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sqrt{{(13^{\frac{1}{2}})}^{2} + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.5.2

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

$\sqrt{13^{\frac{1}{2} \cdot 2} + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.5.3

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

$\sqrt{13^{\frac{2}{2}} + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$\sqrt{13^{\frac{2}{2}} + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.5.4.2

Rewrite the expression.

$\sqrt{13^{1} + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.5.5

Evaluate the exponent.

$\sqrt{13 + {| 3 - 2 i |}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.6

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

$\sqrt{13 + {\sqrt{3^{2} + {(- 2)}^{2}}}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.7

Raise $3$ to the power of $2$ .

$\sqrt{13 + {\sqrt{9 + {(- 2)}^{2}}}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.8

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

$\sqrt{13 + {\sqrt{9 + 4}}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.9

Add $9$ and $4$ .

$\sqrt{13 + {\sqrt{13}}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.10

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sqrt{13 + {(13^{\frac{1}{2}})}^{2} + {| 0 - 1 i |}^{2}}$

Step 2.10.2

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

$\sqrt{13 + 13^{\frac{1}{2} \cdot 2} + {| 0 - 1 i |}^{2}}$

Step 2.10.3

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

$\sqrt{13 + 13^{\frac{2}{2}} + {| 0 - 1 i |}^{2}}$

Step 2.10.4

Cancel the common factor of $2$ .

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Step 2.10.4.1

Cancel the common factor.

$\sqrt{13 + 13^{\frac{2}{2}} + {| 0 - 1 i |}^{2}}$

Step 2.10.4.2

Rewrite the expression.

$\sqrt{13 + 13^{1} + {| 0 - 1 i |}^{2}}$

Step 2.10.5

Evaluate the exponent.

$\sqrt{13 + 13 + {| 0 - 1 i |}^{2}}$

Step 2.11

Rewrite $- 1 i$ as $- i$ .

$\sqrt{13 + 13 + {| 0 - i |}^{2}}$

Step 2.12

Subtract $i$ from $0$ .

$\sqrt{13 + 13 + {| - i |}^{2}}$

Step 2.13

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

$\sqrt{13 + 13 + {\sqrt{0^{2} + {(- 1)}^{2}}}^{2}}$

Step 2.14

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

$\sqrt{13 + 13 + {\sqrt{0 + {(- 1)}^{2}}}^{2}}$

Step 2.15

Raise $- 1$ to the power of $2$ .

$\sqrt{13 + 13 + {\sqrt{0 + 1}}^{2}}$

Step 2.16

Add $0$ and $1$ .

$\sqrt{13 + 13 + {\sqrt{1}}^{2}}$

Step 2.17

Any root of $1$ is $1$ .

$\sqrt{13 + 13 + 1^{2}}$

Step 2.18

One to any power is one.

$\sqrt{13 + 13 + 1}$

Step 2.19

Add $13$ and $13$ .

$\sqrt{26 + 1}$

Step 2.20

Add $26$ and $1$ .

$\sqrt{27}$

Step 2.21

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.21.1

Factor $9$ out of $27$ .

$\sqrt{9 (3)}$

Step 2.21.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} \cdot 3}$

Step 2.22

Pull terms out from under the radical.

$3 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{3}$

Decimal Form:

$5.19615242 \dots$