Linear Algebra Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Multiply $i$ by $1$ .

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

Step 2.2

Add $0$ and $i$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

One to any power is one.

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

Step 2.6

Add $0$ and $1$ .

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

Step 2.7

Any root of $1$ is $1$ .

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

Step 2.8

One to any power is one.

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

Step 2.9

Multiply $i$ by $1$ .

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

Step 2.10

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

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

Step 2.11

One to any power is one.

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

Step 2.12

One to any power is one.

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

Step 2.13

Add $1$ and $1$ .

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

Step 2.14

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

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Step 2.14.1

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

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

Step 2.14.2

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

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

Step 2.14.3

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

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

Step 2.14.4

Cancel the common factor of $2$ .

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Step 2.14.4.1

Cancel the common factor.

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

Step 2.14.4.2

Rewrite the expression.

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

Step 2.14.5

Evaluate the exponent.

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

Step 2.15

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

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

Step 2.16

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

$\sqrt{1 + 2 + {\sqrt{16 + {(- 2)}^{2}}}^{2}}$

Step 2.17

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

$\sqrt{1 + 2 + {\sqrt{16 + 4}}^{2}}$

Step 2.18

Add $16$ and $4$ .

$\sqrt{1 + 2 + {\sqrt{20}}^{2}}$

Step 2.19

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.19.1

Factor $4$ out of $20$ .

$\sqrt{1 + 2 + {\sqrt{4 (5)}}^{2}}$

Step 2.19.2

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

$\sqrt{1 + 2 + {\sqrt{2^{2} \cdot 5}}^{2}}$

Step 2.20

Pull terms out from under the radical.

$\sqrt{1 + 2 + {(2 \sqrt{5})}^{2}}$

Step 2.21

Apply the product rule to $2 \sqrt{5}$ .

$\sqrt{1 + 2 + 2^{2} {\sqrt{5}}^{2}}$

Step 2.22

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

$\sqrt{1 + 2 + 4 {\sqrt{5}}^{2}}$

Step 2.23

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

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Step 2.23.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{1 + 2 + 4 {(5^{\frac{1}{2}})}^{2}}$

Step 2.23.2

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

$\sqrt{1 + 2 + 4 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 2.23.3

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

$\sqrt{1 + 2 + 4 \cdot 5^{\frac{2}{2}}}$

Step 2.23.4

Cancel the common factor of $2$ .

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Step 2.23.4.1

Cancel the common factor.

$\sqrt{1 + 2 + 4 \cdot 5^{\frac{2}{2}}}$

Step 2.23.4.2

Rewrite the expression.

$\sqrt{1 + 2 + 4 \cdot 5^{1}}$

Step 2.23.5

Evaluate the exponent.

$\sqrt{1 + 2 + 4 \cdot 5}$

Step 2.24

Multiply $4$ by $5$ .

$\sqrt{1 + 2 + 20}$

Step 2.25

Add $1$ and $2$ .

$\sqrt{3 + 20}$

Step 2.26

Add $3$ and $20$ .

$\sqrt{23}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{23}$

Decimal Form:

$4.79583152 \dots$