Linear Algebra Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Multiply $0$ by $i$ .

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

Step 2.2

Add $2$ and $0$ .

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

Step 2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Add $4$ and $4$ .

$\sqrt{4 + {\sqrt{8}}^{2} + {| 0 - 3 i |}^{2}}$

Step 2.9

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.9.1

Factor $4$ out of $8$ .

$\sqrt{4 + {\sqrt{4 (2)}}^{2} + {| 0 - 3 i |}^{2}}$

Step 2.9.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.10

Pull terms out from under the radical.

$\sqrt{4 + {(2 \sqrt{2})}^{2} + {| 0 - 3 i |}^{2}}$

Step 2.11

Apply the product rule to $2 \sqrt{2}$ .

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

Step 2.12

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

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

Step 2.13

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

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Step 2.13.1

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

$\sqrt{4 + 4 {(2^{\frac{1}{2}})}^{2} + {| 0 - 3 i |}^{2}}$

Step 2.13.2

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

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

Step 2.13.3

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

$\sqrt{4 + 4 \cdot 2^{\frac{2}{2}} + {| 0 - 3 i |}^{2}}$

Step 2.13.4

Cancel the common factor of $2$ .

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Step 2.13.4.1

Cancel the common factor.

$\sqrt{4 + 4 \cdot 2^{\frac{2}{2}} + {| 0 - 3 i |}^{2}}$

Step 2.13.4.2

Rewrite the expression.

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

Step 2.13.5

Evaluate the exponent.

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

Step 2.14

Multiply $4$ by $2$ .

$\sqrt{4 + 8 + {| 0 - 3 i |}^{2}}$

Step 2.15

Subtract $3 i$ from $0$ .

$\sqrt{4 + 8 + {| - 3 i |}^{2}}$

Step 2.16

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

$\sqrt{4 + 8 + {\sqrt{0^{2} + {(- 3)}^{2}}}^{2}}$

Step 2.17

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

$\sqrt{4 + 8 + {\sqrt{0 + {(- 3)}^{2}}}^{2}}$

Step 2.18

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

$\sqrt{4 + 8 + {\sqrt{0 + 9}}^{2}}$

Step 2.19

Add $0$ and $9$ .

$\sqrt{4 + 8 + {\sqrt{9}}^{2}}$

Step 2.20

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

$\sqrt{4 + 8 + {\sqrt{3^{2}}}^{2}}$

Step 2.21

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{4 + 8 + 3^{2}}$

Step 2.22

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

$\sqrt{4 + 8 + 9}$

Step 2.23

Add $4$ and $8$ .

$\sqrt{12 + 9}$

Step 2.24

Add $12$ and $9$ .

$\sqrt{21}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{21}$

Decimal Form:

$4.58257569 \dots$