Linear Algebra Examples

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

Step 1

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

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

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.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $4$ and $9$ .

$\sqrt{{\sqrt{13}}^{2} + {| 4 - 1 i |}^{2} + {| 3 + 2 i |}^{2}}$

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 $13^{\frac{1}{2}}$ .

$\sqrt{{(13^{\frac{1}{2}})}^{2} + {| 4 - 1 i |}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.5.2

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

$\sqrt{13^{\frac{1}{2} \cdot 2} + {| 4 - 1 i |}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.5.3

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

$\sqrt{13^{\frac{2}{2}} + {| 4 - 1 i |}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$\sqrt{13^{\frac{2}{2}} + {| 4 - 1 i |}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.5.4.2

Rewrite the expression.

$\sqrt{13^{1} + {| 4 - 1 i |}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.5.5

Evaluate the exponent.

$\sqrt{13 + {| 4 - 1 i |}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.6

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

$\sqrt{13 + {| 4 - i |}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.7

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

$\sqrt{13 + {\sqrt{4^{2} + {(- 1)}^{2}}}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.8

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

$\sqrt{13 + {\sqrt{16 + {(- 1)}^{2}}}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.9

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

$\sqrt{13 + {\sqrt{16 + 1}}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.10

Add $16$ and $1$ .

$\sqrt{13 + {\sqrt{17}}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.11

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 2.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\sqrt{13 + {(17^{\frac{1}{2}})}^{2} + {| 3 + 2 i |}^{2}}$

Step 2.11.2

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

$\sqrt{13 + 17^{\frac{1}{2} \cdot 2} + {| 3 + 2 i |}^{2}}$

Step 2.11.3

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

$\sqrt{13 + 17^{\frac{2}{2}} + {| 3 + 2 i |}^{2}}$

Step 2.11.4

Cancel the common factor of $2$ .

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Step 2.11.4.1

Cancel the common factor.

$\sqrt{13 + 17^{\frac{2}{2}} + {| 3 + 2 i |}^{2}}$

Step 2.11.4.2

Rewrite the expression.

$\sqrt{13 + 17^{1} + {| 3 + 2 i |}^{2}}$

Step 2.11.5

Evaluate the exponent.

$\sqrt{13 + 17 + {| 3 + 2 i |}^{2}}$

Step 2.12

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

$\sqrt{13 + 17 + {\sqrt{3^{2} + 2^{2}}}^{2}}$

Step 2.13

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

$\sqrt{13 + 17 + {\sqrt{9 + 2^{2}}}^{2}}$

Step 2.14

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

$\sqrt{13 + 17 + {\sqrt{9 + 4}}^{2}}$

Step 2.15

Add $9$ and $4$ .

$\sqrt{13 + 17 + {\sqrt{13}}^{2}}$

Step 2.16

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

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Step 2.16.1

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

$\sqrt{13 + 17 + {(13^{\frac{1}{2}})}^{2}}$

Step 2.16.2

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

$\sqrt{13 + 17 + 13^{\frac{1}{2} \cdot 2}}$

Step 2.16.3

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

$\sqrt{13 + 17 + 13^{\frac{2}{2}}}$

Step 2.16.4

Cancel the common factor of $2$ .

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Step 2.16.4.1

Cancel the common factor.

$\sqrt{13 + 17 + 13^{\frac{2}{2}}}$

Step 2.16.4.2

Rewrite the expression.

$\sqrt{13 + 17 + 13^{1}}$

Step 2.16.5

Evaluate the exponent.

$\sqrt{13 + 17 + 13}$

Step 2.17

Add $13$ and $17$ .

$\sqrt{30 + 13}$

Step 2.18

Add $30$ and $13$ .

$\sqrt{43}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{43}$

Decimal Form:

$6.55743852 \dots$