Linear Algebra Examples

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

Step 1

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $4$ and $1$ .

$\sqrt{{\sqrt{5}}^{2} + {| 3 + 0 i |}^{2} + {| 1 + 0 i |}^{2}}$

Step 2.6

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

Tap for more steps...

Step 2.6.1

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

$\sqrt{{(5^{\frac{1}{2}})}^{2} + {| 3 + 0 i |}^{2} + {| 1 + 0 i |}^{2}}$

Step 2.6.2

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

$\sqrt{5^{\frac{1}{2} \cdot 2} + {| 3 + 0 i |}^{2} + {| 1 + 0 i |}^{2}}$

Step 2.6.3

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

$\sqrt{5^{\frac{2}{2}} + {| 3 + 0 i |}^{2} + {| 1 + 0 i |}^{2}}$

Step 2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.4.1

Cancel the common factor.

$\sqrt{5^{\frac{2}{2}} + {| 3 + 0 i |}^{2} + {| 1 + 0 i |}^{2}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{5^{1} + {| 3 + 0 i |}^{2} + {| 1 + 0 i |}^{2}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{5 + {| 3 + 0 i |}^{2} + {| 1 + 0 i |}^{2}}$

Step 2.7

Multiply $0$ by $i$ .

$\sqrt{5 + {| 3 + 0 |}^{2} + {| 1 + 0 i |}^{2}}$

Step 2.8

Add $3$ and $0$ .

$\sqrt{5 + {| 3 |}^{2} + {| 1 + 0 i |}^{2}}$

Step 2.9

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\sqrt{5 + 3^{2} + {| 1 + 0 i |}^{2}}$

Step 2.10

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

$\sqrt{5 + 9 + {| 1 + 0 i |}^{2}}$

Step 2.11

Multiply $0$ by $i$ .

$\sqrt{5 + 9 + {| 1 + 0 |}^{2}}$

Step 2.12

Add $1$ and $0$ .

$\sqrt{5 + 9 + {| 1 |}^{2}}$

Step 2.13

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\sqrt{5 + 9 + 1^{2}}$

Step 2.14

One to any power is one.

$\sqrt{5 + 9 + 1}$

Step 2.15

Add $5$ and $9$ .

$\sqrt{14 + 1}$

Step 2.16

Add $14$ and $1$ .

$\sqrt{15}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{15}$

Decimal Form:

$3.87298334 \dots$