Linear Algebra Examples

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

Step 1

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

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

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.

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

Step 2.2

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

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

Step 2.3

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

$\sqrt{{\sqrt{9 + 16}}^{2} + {| 4 + 4 i |}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.4

Add $9$ and $16$ .

$\sqrt{{\sqrt{25}}^{2} + {| 4 + 4 i |}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

$\sqrt{25 + {| 4 + 4 i |}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.8

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

$\sqrt{25 + {\sqrt{4^{2} + 4^{2}}}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.9

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

$\sqrt{25 + {\sqrt{16 + 4^{2}}}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.10

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

$\sqrt{25 + {\sqrt{16 + 16}}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.11

Add $16$ and $16$ .

$\sqrt{25 + {\sqrt{32}}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.12

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.12.1

Factor $16$ out of $32$ .

$\sqrt{25 + {\sqrt{16 (2)}}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.12.2

Rewrite $16$ as $4^{2}$ .

$\sqrt{25 + {\sqrt{4^{2} \cdot 2}}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.13

Pull terms out from under the radical.

$\sqrt{25 + {(4 \sqrt{2})}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.14

Apply the product rule to $4 \sqrt{2}$ .

$\sqrt{25 + 4^{2} {\sqrt{2}}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.15

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

$\sqrt{25 + 16 {\sqrt{2}}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.16

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

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Step 2.16.1

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

$\sqrt{25 + 16 {(2^{\frac{1}{2}})}^{2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.16.2

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

$\sqrt{25 + 16 \cdot 2^{\frac{1}{2} \cdot 2} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.16.3

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

$\sqrt{25 + 16 \cdot 2^{\frac{2}{2}} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.16.4

Cancel the common factor of $2$ .

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Step 2.16.4.1

Cancel the common factor.

$\sqrt{25 + 16 \cdot 2^{\frac{2}{2}} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.16.4.2

Rewrite the expression.

$\sqrt{25 + 16 \cdot 2^{1} + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.16.5

Evaluate the exponent.

$\sqrt{25 + 16 \cdot 2 + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.17

Multiply $16$ by $2$ .

$\sqrt{25 + 32 + {| 2 + 2 i |}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.18

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

$\sqrt{25 + 32 + {\sqrt{2^{2} + 2^{2}}}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.19

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

$\sqrt{25 + 32 + {\sqrt{4 + 2^{2}}}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.20

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

$\sqrt{25 + 32 + {\sqrt{4 + 4}}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.21

Add $4$ and $4$ .

$\sqrt{25 + 32 + {\sqrt{8}}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.22

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

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Step 2.22.1

Factor $4$ out of $8$ .

$\sqrt{25 + 32 + {\sqrt{4 (2)}}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.22.2

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

$\sqrt{25 + 32 + {\sqrt{2^{2} \cdot 2}}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.23

Pull terms out from under the radical.

$\sqrt{25 + 32 + {(2 \sqrt{2})}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.24

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

$\sqrt{25 + 32 + 2^{2} {\sqrt{2}}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.25

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

$\sqrt{25 + 32 + 4 {\sqrt{2}}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.26

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

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Step 2.26.1

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

$\sqrt{25 + 32 + 4 {(2^{\frac{1}{2}})}^{2} + {| 4 - 2 i |}^{2}}$

Step 2.26.2

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

$\sqrt{25 + 32 + 4 \cdot 2^{\frac{1}{2} \cdot 2} + {| 4 - 2 i |}^{2}}$

Step 2.26.3

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

$\sqrt{25 + 32 + 4 \cdot 2^{\frac{2}{2}} + {| 4 - 2 i |}^{2}}$

Step 2.26.4

Cancel the common factor of $2$ .

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Step 2.26.4.1

Cancel the common factor.

$\sqrt{25 + 32 + 4 \cdot 2^{\frac{2}{2}} + {| 4 - 2 i |}^{2}}$

Step 2.26.4.2

Rewrite the expression.

$\sqrt{25 + 32 + 4 \cdot 2^{1} + {| 4 - 2 i |}^{2}}$

Step 2.26.5

Evaluate the exponent.

$\sqrt{25 + 32 + 4 \cdot 2 + {| 4 - 2 i |}^{2}}$

Step 2.27

Multiply $4$ by $2$ .

$\sqrt{25 + 32 + 8 + {| 4 - 2 i |}^{2}}$

Step 2.28

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

$\sqrt{25 + 32 + 8 + {\sqrt{4^{2} + {(- 2)}^{2}}}^{2}}$

Step 2.29

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

$\sqrt{25 + 32 + 8 + {\sqrt{16 + {(- 2)}^{2}}}^{2}}$

Step 2.30

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

$\sqrt{25 + 32 + 8 + {\sqrt{16 + 4}}^{2}}$

Step 2.31

Add $16$ and $4$ .

$\sqrt{25 + 32 + 8 + {\sqrt{20}}^{2}}$

Step 2.32

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.32.1

Factor $4$ out of $20$ .

$\sqrt{25 + 32 + 8 + {\sqrt{4 (5)}}^{2}}$

Step 2.32.2

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

$\sqrt{25 + 32 + 8 + {\sqrt{2^{2} \cdot 5}}^{2}}$

Step 2.33

Pull terms out from under the radical.

$\sqrt{25 + 32 + 8 + {(2 \sqrt{5})}^{2}}$

Step 2.34

Apply the product rule to $2 \sqrt{5}$ .

$\sqrt{25 + 32 + 8 + 2^{2} {\sqrt{5}}^{2}}$

Step 2.35

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

$\sqrt{25 + 32 + 8 + 4 {\sqrt{5}}^{2}}$

Step 2.36

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

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Step 2.36.1

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

$\sqrt{25 + 32 + 8 + 4 {(5^{\frac{1}{2}})}^{2}}$

Step 2.36.2

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

$\sqrt{25 + 32 + 8 + 4 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 2.36.3

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

$\sqrt{25 + 32 + 8 + 4 \cdot 5^{\frac{2}{2}}}$

Step 2.36.4

Cancel the common factor of $2$ .

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Step 2.36.4.1

Cancel the common factor.

$\sqrt{25 + 32 + 8 + 4 \cdot 5^{\frac{2}{2}}}$

Step 2.36.4.2

Rewrite the expression.

$\sqrt{25 + 32 + 8 + 4 \cdot 5^{1}}$

Step 2.36.5

Evaluate the exponent.

$\sqrt{25 + 32 + 8 + 4 \cdot 5}$

Step 2.37

Multiply $4$ by $5$ .

$\sqrt{25 + 32 + 8 + 20}$

Step 2.38

Add $25$ and $32$ .

$\sqrt{57 + 8 + 20}$

Step 2.39

Add $57$ and $8$ .

$\sqrt{65 + 20}$

Step 2.40

Add $65$ and $20$ .

$\sqrt{85}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{85}$

Decimal Form:

$9.21954445 \dots$