Linear Algebra Examples

$[\begin{array}{c} 4 + 0 i \\ 0 + 2 i \\ 1 - 1 i \\ 4 - 3 i \end{array}]$

Step 1

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

$\sqrt{{| 4 + 0 i |}^{2} + {| 0 + 2 i |}^{2} + {| 1 - 1 i |}^{2} + {| 4 - 3 i |}^{2}}$

Step 2

Simplify.

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Step 2.1

Multiply $0$ by $i$ .

$\sqrt{{| 4 + 0 |}^{2} + {| 0 + 2 i |}^{2} + {| 1 - 1 i |}^{2} + {| 4 - 3 i |}^{2}}$

Step 2.2

Add $4$ and $0$ .

$\sqrt{{| 4 |}^{2} + {| 0 + 2 i |}^{2} + {| 1 - 1 i |}^{2} + {| 4 - 3 i |}^{2}}$

Step 2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\sqrt{4^{2} + {| 0 + 2 i |}^{2} + {| 1 - 1 i |}^{2} + {| 4 - 3 i |}^{2}}$

Step 2.4

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

$\sqrt{16 + {| 0 + 2 i |}^{2} + {| 1 - 1 i |}^{2} + {| 4 - 3 i |}^{2}}$

Step 2.5

Add $0$ and $2 i$ .

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

Step 2.6

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

$\sqrt{16 + {\sqrt{0^{2} + 2^{2}}}^{2} + {| 1 - 1 i |}^{2} + {| 4 - 3 i |}^{2}}$

Step 2.7

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

$\sqrt{16 + {\sqrt{0 + 2^{2}}}^{2} + {| 1 - 1 i |}^{2} + {| 4 - 3 i |}^{2}}$

Step 2.8

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

$\sqrt{16 + {\sqrt{0 + 4}}^{2} + {| 1 - 1 i |}^{2} + {| 4 - 3 i |}^{2}}$

Step 2.9

Add $0$ and $4$ .

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

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

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

Step 2.15

One to any power is one.

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

Step 2.16

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

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

Step 2.17

Add $1$ and $1$ .

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

Step 2.18

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

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Step 2.18.1

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

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

Step 2.18.2

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

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

Step 2.18.3

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

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

Step 2.18.4

Cancel the common factor of $2$ .

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Step 2.18.4.1

Cancel the common factor.

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

Step 2.18.4.2

Rewrite the expression.

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

Step 2.18.5

Evaluate the exponent.

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

Step 2.19

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

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

Step 2.20

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

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

Step 2.21

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

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

Step 2.22

Add $16$ and $9$ .

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

Step 2.23

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

$\sqrt{16 + 4 + 2 + {\sqrt{5^{2}}}^{2}}$

Step 2.24

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

$\sqrt{16 + 4 + 2 + 5^{2}}$

Step 2.25

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

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

Step 2.26

Add $16$ and $4$ .

$\sqrt{20 + 2 + 25}$

Step 2.27

Add $20$ and $2$ .

$\sqrt{22 + 25}$

Step 2.28

Add $22$ and $25$ .

$\sqrt{47}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{47}$

Decimal Form:

$6.85565460 \dots$