Linear Algebra Examples

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

Step 1

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $16$ and $4$ .

$\sqrt{{\sqrt{20}}^{2} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.5

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

Tap for more steps...

Step 2.5.1

Factor $4$ out of $20$ .

$\sqrt{{\sqrt{4 (5)}}^{2} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.5.2

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

$\sqrt{{\sqrt{2^{2} \cdot 5}}^{2} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.6

Pull terms out from under the radical.

$\sqrt{{(2 \sqrt{5})}^{2} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.7

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

$\sqrt{2^{2} {\sqrt{5}}^{2} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.8

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

$\sqrt{4 {\sqrt{5}}^{2} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.9

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

Tap for more steps...

Step 2.9.1

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

$\sqrt{4 {(5^{\frac{1}{2}})}^{2} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.9.2

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

$\sqrt{4 \cdot 5^{\frac{1}{2} \cdot 2} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.9.3

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

$\sqrt{4 \cdot 5^{\frac{2}{2}} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.9.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.9.4.1

Cancel the common factor.

$\sqrt{4 \cdot 5^{\frac{2}{2}} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.9.4.2

Rewrite the expression.

$\sqrt{4 \cdot 5^{1} + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.9.5

Evaluate the exponent.

$\sqrt{4 \cdot 5 + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.10

Multiply $4$ by $5$ .

$\sqrt{20 + {| 4 + 0 i |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.11

Multiply $0$ by $i$ .

$\sqrt{20 + {| 4 + 0 |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.12

Add $4$ and $0$ .

$\sqrt{20 + {| 4 |}^{2} + {| 1 - 3 i |}^{2}}$

Step 2.13

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

$\sqrt{20 + 4^{2} + {| 1 - 3 i |}^{2}}$

Step 2.14

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

$\sqrt{20 + 16 + {| 1 - 3 i |}^{2}}$

Step 2.15

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

$\sqrt{20 + 16 + {\sqrt{1^{2} + {(- 3)}^{2}}}^{2}}$

Step 2.16

One to any power is one.

$\sqrt{20 + 16 + {\sqrt{1 + {(- 3)}^{2}}}^{2}}$

Step 2.17

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

$\sqrt{20 + 16 + {\sqrt{1 + 9}}^{2}}$

Step 2.18

Add $1$ and $9$ .

$\sqrt{20 + 16 + {\sqrt{10}}^{2}}$

Step 2.19

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Step 2.19.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{20 + 16 + {(10^{\frac{1}{2}})}^{2}}$

Step 2.19.2

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

$\sqrt{20 + 16 + 10^{\frac{1}{2} \cdot 2}}$

Step 2.19.3

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

$\sqrt{20 + 16 + 10^{\frac{2}{2}}}$

Step 2.19.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.19.4.1

Cancel the common factor.

$\sqrt{20 + 16 + 10^{\frac{2}{2}}}$

Step 2.19.4.2

Rewrite the expression.

$\sqrt{20 + 16 + 10^{1}}$

Step 2.19.5

Evaluate the exponent.

$\sqrt{20 + 16 + 10}$

Step 2.20

Add $20$ and $16$ .

$\sqrt{36 + 10}$

Step 2.21

Add $36$ and $10$ .

$\sqrt{46}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{46}$

Decimal Form:

$6.78232998 \dots$