Linear Algebra Examples

$[\begin{array}{c} 4 \\ 0 \\ 4 \\ 5 \\ 6 \\ 7 \\ 8 \end{array}]$

Step 1

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

$\sqrt{4^{2} + 0^{2} + 4^{2} + 5^{2} + 6^{2} + 7^{2} + 8^{2}}$

Step 2

Simplify.

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Step 2.1

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

$\sqrt{16 + 0^{2} + 4^{2} + 5^{2} + 6^{2} + 7^{2} + 8^{2}}$

Step 2.2

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

$\sqrt{16 + 0 + 4^{2} + 5^{2} + 6^{2} + 7^{2} + 8^{2}}$

Step 2.3

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

$\sqrt{16 + 0 + 16 + 5^{2} + 6^{2} + 7^{2} + 8^{2}}$

Step 2.4

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

$\sqrt{16 + 0 + 16 + 25 + 6^{2} + 7^{2} + 8^{2}}$

Step 2.5

Raise $6$ to the power of $2$ .

$\sqrt{16 + 0 + 16 + 25 + 36 + 7^{2} + 8^{2}}$

Step 2.6

Raise $7$ to the power of $2$ .

$\sqrt{16 + 0 + 16 + 25 + 36 + 49 + 8^{2}}$

Step 2.7

Raise $8$ to the power of $2$ .

$\sqrt{16 + 0 + 16 + 25 + 36 + 49 + 64}$

Step 2.8

Add $16$ and $0$ .

$\sqrt{16 + 16 + 25 + 36 + 49 + 64}$

Step 2.9

Add $16$ and $16$ .

$\sqrt{32 + 25 + 36 + 49 + 64}$

Step 2.10

Add $32$ and $25$ .

$\sqrt{57 + 36 + 49 + 64}$

Step 2.11

Add $57$ and $36$ .

$\sqrt{93 + 49 + 64}$

Step 2.12

Add $93$ and $49$ .

$\sqrt{142 + 64}$

Step 2.13

Add $142$ and $64$ .

$\sqrt{206}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{206}$

Decimal Form:

$14.35270009 \dots$