Linear Algebra Examples

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 4 - 1 i 4 - 2 i 2 + 2 i 3 - 3 i \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

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}}

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

Step 2

Simplify.

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Step 2.1

Rewrite

- 1 i

$- 1 i$ as

- i

$- i$ .

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

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

Step 2.2

Use the formula

| a + b i | = \sqrt{a^{2} + b^{2}}

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

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

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

Step 2.3

Raise

4

$4$ to the power of

2

$2$ .

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

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

Step 2.4

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 2.5

Add

16

$16$ and

1

$1$ .

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

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

Step 2.6

Rewrite

{\sqrt{17}}^{2}

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

17

$17$ .

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Step 2.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{17}

$\sqrt{17}$ as

17^{\frac{1}{2}}

$17^{\frac{1}{2}}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

$7.93725393 \dots$