Linear Algebra Examples

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

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

Step 1

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

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

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

Step 2

Simplify.

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Step 2.1

Multiply

0

$0$ by

i

$i$ .

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

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

Step 2.2

Add

2

$2$ and

0

$0$ .

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

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

Step 2.3

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

2

$2$ is

2

$2$ .

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

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

Step 2.4

Raise

2

$2$ to the power of

2

$2$ .

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

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

Step 2.5

Use the formula

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

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

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

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

Step 2.6

Raise

3

$3$ to the power of

2

$2$ .

\sqrt{4 + {\sqrt{9 + {(- 4)}^{2}}}^{2} + {| 4 + 3 i |}^{2}}

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

Step 2.7

Raise

- 4

$- 4$ to the power of

2

$2$ .

\sqrt{4 + {\sqrt{9 + 16}}^{2} + {| 4 + 3 i |}^{2}}

$\sqrt{4 + {\sqrt{9 + 16}}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.8

Add

9

$9$ and

16

$16$ .

\sqrt{4 + {\sqrt{25}}^{2} + {| 4 + 3 i |}^{2}}

$\sqrt{4 + {\sqrt{25}}^{2} + {| 4 + 3 i |}^{2}}$

Step 2.9

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

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

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

Step 2.10

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

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

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

Step 2.11

Raise

5

$5$ to the power of

2

$2$ .

\sqrt{4 + 25 + {| 4 + 3 i |}^{2}}

$\sqrt{4 + 25 + {| 4 + 3 i |}^{2}}$

Step 2.12

Use the formula

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

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

\sqrt{4 + 25 + {\sqrt{4^{2} + 3^{2}}}^{2}}

$\sqrt{4 + 25 + {\sqrt{4^{2} + 3^{2}}}^{2}}$

Step 2.13

Raise

4

$4$ to the power of

2

$2$ .

\sqrt{4 + 25 + {\sqrt{16 + 3^{2}}}^{2}}

$\sqrt{4 + 25 + {\sqrt{16 + 3^{2}}}^{2}}$

Step 2.14

Raise

3

$3$ to the power of

2

$2$ .

\sqrt{4 + 25 + {\sqrt{16 + 9}}^{2}}

$\sqrt{4 + 25 + {\sqrt{16 + 9}}^{2}}$

Step 2.15

Add

16

$16$ and

9

$9$ .

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

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

Step 2.16

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

\sqrt{4 + 25 + {\sqrt{5^{2}}}^{2}}

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

Step 2.17

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

\sqrt{4 + 25 + 5^{2}}

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

Step 2.18

Raise

5

$5$ to the power of

2

$2$ .

\sqrt{4 + 25 + 25}

$\sqrt{4 + 25 + 25}$

Step 2.19

Add

4

$4$ and

25

$25$ .

\sqrt{29 + 25}

$\sqrt{29 + 25}$

Step 2.20

Add

29

$29$ and

25

$25$ .

\sqrt{54}

$\sqrt{54}$

Step 2.21

Rewrite

54

$54$ as

3^{2} \cdot 6

$3^{2} \cdot 6$ .

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Step 2.21.1

Factor

9

$9$ out of

54

$54$ .

\sqrt{9 (6)}

$\sqrt{9 (6)}$

Step 2.21.2

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

\sqrt{3^{2} \cdot 6}

$\sqrt{3^{2} \cdot 6}$

\sqrt{3^{2} \cdot 6}

$\sqrt{3^{2} \cdot 6}$

Step 2.22

Pull terms out from under the radical.

3 \sqrt{6}

$3 \sqrt{6}$

3 \sqrt{6}

$3 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

3 \sqrt{6}

$3 \sqrt{6}$

Decimal Form:

7.34846922 \dots

$7.34846922 \dots$