Linear Algebra Examples

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

Step 1

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Add $0$ and $3 i$ .

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

Step 2.2

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

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

Step 2.3

Raising $0$ to any positive power yields $0$ .

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

Step 2.4

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

$\sqrt{{\sqrt{0 + 9}}^{2} + {| 4 + 3 i |}^{2} + {| 2 + 0 i |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.5

Add $0$ and $9$ .

$\sqrt{{\sqrt{9}}^{2} + {| 4 + 3 i |}^{2} + {| 2 + 0 i |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

$\sqrt{9 + {| 4 + 3 i |}^{2} + {| 2 + 0 i |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.9

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

$\sqrt{9 + {\sqrt{4^{2} + 3^{2}}}^{2} + {| 2 + 0 i |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.10

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

$\sqrt{9 + {\sqrt{16 + 3^{2}}}^{2} + {| 2 + 0 i |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.11

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

$\sqrt{9 + {\sqrt{16 + 9}}^{2} + {| 2 + 0 i |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.12

Add $16$ and $9$ .

$\sqrt{9 + {\sqrt{25}}^{2} + {| 2 + 0 i |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.13

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

$\sqrt{9 + {\sqrt{5^{2}}}^{2} + {| 2 + 0 i |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.14

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

$\sqrt{9 + 5^{2} + {| 2 + 0 i |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.15

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

$\sqrt{9 + 25 + {| 2 + 0 i |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.16

Multiply $0$ by $i$ .

$\sqrt{9 + 25 + {| 2 + 0 |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.17

Add $2$ and $0$ .

$\sqrt{9 + 25 + {| 2 |}^{2} + {| 2 - 3 i |}^{2}}$

Step 2.18

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\sqrt{9 + 25 + 2^{2} + {| 2 - 3 i |}^{2}}$

Step 2.19

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

$\sqrt{9 + 25 + 4 + {| 2 - 3 i |}^{2}}$

Step 2.20

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

$\sqrt{9 + 25 + 4 + {\sqrt{2^{2} + {(- 3)}^{2}}}^{2}}$

Step 2.21

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

$\sqrt{9 + 25 + 4 + {\sqrt{4 + {(- 3)}^{2}}}^{2}}$

Step 2.22

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

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

Step 2.23

Add $4$ and $9$ .

$\sqrt{9 + 25 + 4 + {\sqrt{13}}^{2}}$

Step 2.24

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Tap for more steps...

Step 2.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sqrt{9 + 25 + 4 + {(13^{\frac{1}{2}})}^{2}}$

Step 2.24.2

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

$\sqrt{9 + 25 + 4 + 13^{\frac{1}{2} \cdot 2}}$

Step 2.24.3

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

$\sqrt{9 + 25 + 4 + 13^{\frac{2}{2}}}$

Step 2.24.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.24.4.1

Cancel the common factor.

$\sqrt{9 + 25 + 4 + 13^{\frac{2}{2}}}$

Step 2.24.4.2

Rewrite the expression.

$\sqrt{9 + 25 + 4 + 13^{1}}$

Step 2.24.5

Evaluate the exponent.

$\sqrt{9 + 25 + 4 + 13}$

Step 2.25

Add $9$ and $25$ .

$\sqrt{34 + 4 + 13}$

Step 2.26

Add $34$ and $4$ .

$\sqrt{38 + 13}$

Step 2.27

Add $38$ and $13$ .

$\sqrt{51}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{51}$

Decimal Form:

$7.14142842 \dots$