Linear Algebra Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$\sqrt{{\sqrt{16 + {(- 1)}^{2}}}^{2} + {| 4 - 2 i |}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.4

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

$\sqrt{{\sqrt{16 + 1}}^{2} + {| 4 - 2 i |}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.5

Add $16$ and $1$ .

$\sqrt{{\sqrt{17}}^{2} + {| 4 - 2 i |}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.6

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

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Step 2.6.1

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

$\sqrt{{(17^{\frac{1}{2}})}^{2} + {| 4 - 2 i |}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.6.2

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

$\sqrt{17^{\frac{1}{2} \cdot 2} + {| 4 - 2 i |}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.6.3

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

$\sqrt{17^{\frac{2}{2}} + {| 4 - 2 i |}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{17^{\frac{2}{2}} + {| 4 - 2 i |}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{17^{1} + {| 4 - 2 i |}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{17 + {| 4 - 2 i |}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.7

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

$\sqrt{17 + {\sqrt{4^{2} + {(- 2)}^{2}}}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.8

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

$\sqrt{17 + {\sqrt{16 + {(- 2)}^{2}}}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.9

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

$\sqrt{17 + {\sqrt{16 + 4}}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.10

Add $16$ and $4$ .

$\sqrt{17 + {\sqrt{20}}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.11

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

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Step 2.11.1

Factor $4$ out of $20$ .

$\sqrt{17 + {\sqrt{4 (5)}}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.11.2

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

$\sqrt{17 + {\sqrt{2^{2} \cdot 5}}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.12

Pull terms out from under the radical.

$\sqrt{17 + {(2 \sqrt{5})}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.13

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

$\sqrt{17 + 2^{2} {\sqrt{5}}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.14

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

$\sqrt{17 + 4 {\sqrt{5}}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.15

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

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Step 2.15.1

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

$\sqrt{17 + 4 {(5^{\frac{1}{2}})}^{2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.15.2

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

$\sqrt{17 + 4 \cdot 5^{\frac{1}{2} \cdot 2} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.15.3

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

$\sqrt{17 + 4 \cdot 5^{\frac{2}{2}} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.15.4

Cancel the common factor of $2$ .

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Step 2.15.4.1

Cancel the common factor.

$\sqrt{17 + 4 \cdot 5^{\frac{2}{2}} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.15.4.2

Rewrite the expression.

$\sqrt{17 + 4 \cdot 5^{1} + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.15.5

Evaluate the exponent.

$\sqrt{17 + 4 \cdot 5 + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.16

Multiply $4$ by $5$ .

$\sqrt{17 + 20 + {| 2 + 2 i |}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.17

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

$\sqrt{17 + 20 + {\sqrt{2^{2} + 2^{2}}}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.18

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

$\sqrt{17 + 20 + {\sqrt{4 + 2^{2}}}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.19

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

$\sqrt{17 + 20 + {\sqrt{4 + 4}}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.20

Add $4$ and $4$ .

$\sqrt{17 + 20 + {\sqrt{8}}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.21

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

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Step 2.21.1

Factor $4$ out of $8$ .

$\sqrt{17 + 20 + {\sqrt{4 (2)}}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.21.2

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

$\sqrt{17 + 20 + {\sqrt{2^{2} \cdot 2}}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.22

Pull terms out from under the radical.

$\sqrt{17 + 20 + {(2 \sqrt{2})}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.23

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

$\sqrt{17 + 20 + 2^{2} {\sqrt{2}}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.24

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

$\sqrt{17 + 20 + 4 {\sqrt{2}}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.25

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

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Step 2.25.1

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

$\sqrt{17 + 20 + 4 {(2^{\frac{1}{2}})}^{2} + {| 3 - 3 i |}^{2}}$

Step 2.25.2

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

$\sqrt{17 + 20 + 4 \cdot 2^{\frac{1}{2} \cdot 2} + {| 3 - 3 i |}^{2}}$

Step 2.25.3

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

$\sqrt{17 + 20 + 4 \cdot 2^{\frac{2}{2}} + {| 3 - 3 i |}^{2}}$

Step 2.25.4

Cancel the common factor of $2$ .

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Step 2.25.4.1

Cancel the common factor.

$\sqrt{17 + 20 + 4 \cdot 2^{\frac{2}{2}} + {| 3 - 3 i |}^{2}}$

Step 2.25.4.2

Rewrite the expression.

$\sqrt{17 + 20 + 4 \cdot 2^{1} + {| 3 - 3 i |}^{2}}$

Step 2.25.5

Evaluate the exponent.

$\sqrt{17 + 20 + 4 \cdot 2 + {| 3 - 3 i |}^{2}}$

Step 2.26

Multiply $4$ by $2$ .

$\sqrt{17 + 20 + 8 + {| 3 - 3 i |}^{2}}$

Step 2.27

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

$\sqrt{17 + 20 + 8 + {\sqrt{3^{2} + {(- 3)}^{2}}}^{2}}$

Step 2.28

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

$\sqrt{17 + 20 + 8 + {\sqrt{9 + {(- 3)}^{2}}}^{2}}$

Step 2.29

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

$\sqrt{17 + 20 + 8 + {\sqrt{9 + 9}}^{2}}$

Step 2.30

Add $9$ and $9$ .

$\sqrt{17 + 20 + 8 + {\sqrt{18}}^{2}}$

Step 2.31

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.31.1

Factor $9$ out of $18$ .

$\sqrt{17 + 20 + 8 + {\sqrt{9 (2)}}^{2}}$

Step 2.31.2

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

$\sqrt{17 + 20 + 8 + {\sqrt{3^{2} \cdot 2}}^{2}}$

Step 2.32

Pull terms out from under the radical.

$\sqrt{17 + 20 + 8 + {(3 \sqrt{2})}^{2}}$

Step 2.33

Apply the product rule to $3 \sqrt{2}$ .

$\sqrt{17 + 20 + 8 + 3^{2} {\sqrt{2}}^{2}}$

Step 2.34

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

$\sqrt{17 + 20 + 8 + 9 {\sqrt{2}}^{2}}$

Step 2.35

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

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Step 2.35.1

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

$\sqrt{17 + 20 + 8 + 9 {(2^{\frac{1}{2}})}^{2}}$

Step 2.35.2

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

$\sqrt{17 + 20 + 8 + 9 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 2.35.3

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

$\sqrt{17 + 20 + 8 + 9 \cdot 2^{\frac{2}{2}}}$

Step 2.35.4

Cancel the common factor of $2$ .

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Step 2.35.4.1

Cancel the common factor.

$\sqrt{17 + 20 + 8 + 9 \cdot 2^{\frac{2}{2}}}$

Step 2.35.4.2

Rewrite the expression.

$\sqrt{17 + 20 + 8 + 9 \cdot 2^{1}}$

Step 2.35.5

Evaluate the exponent.

$\sqrt{17 + 20 + 8 + 9 \cdot 2}$

Step 2.36

Multiply $9$ by $2$ .

$\sqrt{17 + 20 + 8 + 18}$

Step 2.37

Add $17$ and $20$ .

$\sqrt{37 + 8 + 18}$

Step 2.38

Add $37$ and $8$ .

$\sqrt{45 + 18}$

Step 2.39

Add $45$ and $18$ .

$\sqrt{63}$

Step 2.40

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 2.40.1

Factor $9$ out of $63$ .

$\sqrt{9 (7)}$

Step 2.40.2

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

$\sqrt{3^{2} \cdot 7}$

Step 2.41

Pull terms out from under the radical.

$3 \sqrt{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{7}$

Decimal Form:

$7.93725393 \dots$