Linear Algebra Examples

$[\begin{array}{c} \frac{14}{5} \\ \frac{2}{5} \end{array}]$

Step 1

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

$\sqrt{{(\frac{14}{5})}^{2} + {(\frac{2}{5})}^{2}}$

Step 2

Simplify.

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Step 2.1

Apply the product rule to $\frac{14}{5}$ .

$\sqrt{\frac{14^{2}}{5^{2}} + {(\frac{2}{5})}^{2}}$

Step 2.2

Raise $14$ to the power of $2$ .

$\sqrt{\frac{196}{5^{2}} + {(\frac{2}{5})}^{2}}$

Step 2.3

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

$\sqrt{\frac{196}{25} + {(\frac{2}{5})}^{2}}$

Step 2.4

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

$\sqrt{\frac{196}{25} + \frac{2^{2}}{5^{2}}}$

Step 2.5

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

$\sqrt{\frac{196}{25} + \frac{4}{5^{2}}}$

Step 2.6

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

$\sqrt{\frac{196}{25} + \frac{4}{25}}$

Step 2.7

Combine the numerators over the common denominator.

$\sqrt{\frac{196 + 4}{25}}$

Step 2.8

Add $196$ and $4$ .

$\sqrt{\frac{200}{25}}$

Step 2.9

Divide $200$ by $25$ .

$\sqrt{8}$

Step 2.10

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.10.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)}$

Step 2.10.2

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

$\sqrt{2^{2} \cdot 2}$

Step 2.11

Pull terms out from under the radical.

$2 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2}$

Decimal Form:

$2.82842712 \dots$