Linear Algebra Examples

$[\begin{array}{c} 2 - 1 i \\ 3 + 1 i \\ 3 + 4 i \end{array}]$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $4$ and $1$ .

$\sqrt{{\sqrt{5}}^{2} + {| 3 + 1 i |}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.6

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

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Step 2.6.1

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

$\sqrt{{(5^{\frac{1}{2}})}^{2} + {| 3 + 1 i |}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.6.2

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

$\sqrt{5^{\frac{1}{2} \cdot 2} + {| 3 + 1 i |}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.6.3

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

$\sqrt{5^{\frac{2}{2}} + {| 3 + 1 i |}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{5^{\frac{2}{2}} + {| 3 + 1 i |}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{5^{1} + {| 3 + 1 i |}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{5 + {| 3 + 1 i |}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.7

Multiply $i$ by $1$ .

$\sqrt{5 + {| 3 + i |}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.8

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

$\sqrt{5 + {\sqrt{3^{2} + 1^{2}}}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.9

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

$\sqrt{5 + {\sqrt{9 + 1^{2}}}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.10

One to any power is one.

$\sqrt{5 + {\sqrt{9 + 1}}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.11

Add $9$ and $1$ .

$\sqrt{5 + {\sqrt{10}}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.12

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.12.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{5 + {(10^{\frac{1}{2}})}^{2} + {| 3 + 4 i |}^{2}}$

Step 2.12.2

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

$\sqrt{5 + 10^{\frac{1}{2} \cdot 2} + {| 3 + 4 i |}^{2}}$

Step 2.12.3

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

$\sqrt{5 + 10^{\frac{2}{2}} + {| 3 + 4 i |}^{2}}$

Step 2.12.4

Cancel the common factor of $2$ .

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Step 2.12.4.1

Cancel the common factor.

$\sqrt{5 + 10^{\frac{2}{2}} + {| 3 + 4 i |}^{2}}$

Step 2.12.4.2

Rewrite the expression.

$\sqrt{5 + 10^{1} + {| 3 + 4 i |}^{2}}$

Step 2.12.5

Evaluate the exponent.

$\sqrt{5 + 10 + {| 3 + 4 i |}^{2}}$

Step 2.13

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

$\sqrt{5 + 10 + {\sqrt{3^{2} + 4^{2}}}^{2}}$

Step 2.14

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

$\sqrt{5 + 10 + {\sqrt{9 + 4^{2}}}^{2}}$

Step 2.15

Raise $4$ to the power of $2$ .

$\sqrt{5 + 10 + {\sqrt{9 + 16}}^{2}}$

Step 2.16

Add $9$ and $16$ .

$\sqrt{5 + 10 + {\sqrt{25}}^{2}}$

Step 2.17

Rewrite $25$ as $5^{2}$ .

$\sqrt{5 + 10 + {\sqrt{5^{2}}}^{2}}$

Step 2.18

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

$\sqrt{5 + 10 + 5^{2}}$

Step 2.19

Raise $5$ to the power of $2$ .

$\sqrt{5 + 10 + 25}$

Step 2.20

Add $5$ and $10$ .

$\sqrt{15 + 25}$

Step 2.21

Add $15$ and $25$ .

$\sqrt{40}$

Step 2.22

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 2.22.1

Factor $4$ out of $40$ .

$\sqrt{4 (10)}$

Step 2.22.2

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

$\sqrt{2^{2} \cdot 10}$

Step 2.23

Pull terms out from under the radical.

$2 \sqrt{10}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{10}$

Decimal Form:

$6.32455532 \dots$