Linear Algebra Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

One to any power is one.

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

Step 2.3

One to any power is one.

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

Step 2.4

Add $1$ and $1$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Evaluate the exponent.

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

Step 2.6

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

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

Step 2.7

One to any power is one.

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

Step 2.8

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

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

Step 2.9

Add $1$ and $1$ .

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

Step 2.10

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

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Step 2.10.1

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

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

Step 2.10.2

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

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

Step 2.10.3

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

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

Step 2.10.4

Cancel the common factor of $2$ .

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Step 2.10.4.1

Cancel the common factor.

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

Step 2.10.4.2

Rewrite the expression.

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

Step 2.10.5

Evaluate the exponent.

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

Step 2.11

One to any power is one.

$\sqrt{2 + 2 + 1}$

Step 2.12

Add $2$ and $2$ .

$\sqrt{4 + 1}$

Step 2.13

Add $4$ and $1$ .

$\sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{5}$

Decimal Form:

$2.23606797 \dots$