Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 4 + 3 i 4 - 2 i 0 - 4 i \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 4 + 3 i \\ 4 - 2 i \\ 0 - 4 i \end{array}]$

Step 1

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

\sqrt{{| 4 + 3 i |}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{{| 4 + 3 i |}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2

Simplify.

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Step 2.1

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} + 3^{2}}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{{\sqrt{4^{2} + 3^{2}}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.2

Raise

4

$4$ to the power of

2

$2$ .

\sqrt{{\sqrt{16 + 3^{2}}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{{\sqrt{16 + 3^{2}}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.3

Raise

3

$3$ to the power of

2

$2$ .

\sqrt{{\sqrt{16 + 9}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{{\sqrt{16 + 9}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.4

Add

16

$16$ and

9

$9$ .

\sqrt{{\sqrt{25}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{{\sqrt{25}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.5

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

\sqrt{{\sqrt{5^{2}}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{{\sqrt{5^{2}}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.6

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

\sqrt{5^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{5^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.7

Raise

5

$5$ to the power of

2

$2$ .

\sqrt{25 + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{25 + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.8

Use the formula

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

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

\sqrt{25 + {\sqrt{4^{2} + {(- 2)}^{2}}}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{25 + {\sqrt{4^{2} + {(- 2)}^{2}}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.9

Raise

4

$4$ to the power of

2

$2$ .

\sqrt{25 + {\sqrt{16 + {(- 2)}^{2}}}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{25 + {\sqrt{16 + {(- 2)}^{2}}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.10

Raise

- 2

$- 2$ to the power of

2

$2$ .

\sqrt{25 + {\sqrt{16 + 4}}^{2} + {| 0 - 4 i |}^{2}}

$\sqrt{25 + {\sqrt{16 + 4}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.11

Add

$16$ and

$4$ .

$\sqrt{25 + {\sqrt{20}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.12

Rewrite

$20$ as

$2^{2} \cdot 5$ .

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Step 2.12.1

Factor

$4$ out of

$20$ .

$\sqrt{25 + {\sqrt{4 (5)}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.12.2

Rewrite

$4$ as

$2^{2}$ .

$\sqrt{25 + {\sqrt{2^{2} \cdot 5}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.13

Pull terms out from under the radical.

$\sqrt{25 + {(2 \sqrt{5})}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.14

Apply the product rule to

$2 \sqrt{5}$ .

$\sqrt{25 + 2^{2} {\sqrt{5}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.15

Raise

$2$ to the power of

$2$ .

$\sqrt{25 + 4 {\sqrt{5}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.16

Rewrite

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

$5$ .

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Step 2.16.1

Use

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

$\sqrt{5}$ as

$5^{\frac{1}{2}}$ .

$\sqrt{25 + 4 {(5^{\frac{1}{2}})}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.16.2

Apply the power rule and multiply exponents,

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

$\sqrt{25 + 4 \cdot 5^{\frac{1}{2} \cdot 2} + {| 0 - 4 i |}^{2}}$

Step 2.16.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{25 + 4 \cdot 5^{\frac{2}{2}} + {| 0 - 4 i |}^{2}}$

Step 2.16.4

Cancel the common factor of

$2$ .

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Step 2.16.4.1

Cancel the common factor.

$\sqrt{25 + 4 \cdot 5^{\frac{2}{2}} + {| 0 - 4 i |}^{2}}$

Step 2.16.4.2

Rewrite the expression.

$\sqrt{25 + 4 \cdot 5^{1} + {| 0 - 4 i |}^{2}}$

Step 2.16.5

Evaluate the exponent.

$\sqrt{25 + 4 \cdot 5 + {| 0 - 4 i |}^{2}}$

Step 2.17

Multiply

$4$ by

$5$ .

$\sqrt{25 + 20 + {| 0 - 4 i |}^{2}}$

Step 2.18

Subtract

$4 i$ from

$0$ .

$\sqrt{25 + 20 + {| - 4 i |}^{2}}$

Step 2.19

Use the formula

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

$\sqrt{25 + 20 + {\sqrt{0^{2} + {(- 4)}^{2}}}^{2}}$

Step 2.20

Raising

$0$ to any positive power yields

$0$ .

$\sqrt{25 + 20 + {\sqrt{0 + {(- 4)}^{2}}}^{2}}$

Step 2.21

Raise

$- 4$ to the power of

$2$ .

$\sqrt{25 + 20 + {\sqrt{0 + 16}}^{2}}$

Step 2.22

Add

$0$ and

$16$ .

$\sqrt{25 + 20 + {\sqrt{16}}^{2}}$

Step 2.23

Rewrite

$16$ as

$4^{2}$ .

$\sqrt{25 + 20 + {\sqrt{4^{2}}}^{2}}$

Step 2.24

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

$\sqrt{25 + 20 + 4^{2}}$

Step 2.25

Raise

$4$ to the power of

$2$ .

$\sqrt{25 + 20 + 16}$

Step 2.26

Add

$25$ and

$20$ .

$\sqrt{45 + 16}$

Step 2.27

Add

$45$ and

$16$ .

$\sqrt{61}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{61}$

Decimal Form:

$7.81024967 \dots$