Linear Algebra Examples

$[\begin{array}{c} 1 & - 1 & 2 & 3 \\ 0 & 1 & 4 & 3 \\ 1 & 0 & 3 & 5 \end{array}]$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

One to any power is one.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$\sqrt{1 + 1 + 4 + 9 + 0^{2} + 1^{2} + 4^{2} + 3^{2} + 1^{2} + 0^{2} + 3^{2} + 5^{2}}$

Step 2.5

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

$\sqrt{1 + 1 + 4 + 9 + 0 + 1^{2} + 4^{2} + 3^{2} + 1^{2} + 0^{2} + 3^{2} + 5^{2}}$

Step 2.6

One to any power is one.

$\sqrt{1 + 1 + 4 + 9 + 0 + 1 + 4^{2} + 3^{2} + 1^{2} + 0^{2} + 3^{2} + 5^{2}}$

Step 2.7

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

$\sqrt{1 + 1 + 4 + 9 + 0 + 1 + 16 + 3^{2} + 1^{2} + 0^{2} + 3^{2} + 5^{2}}$

Step 2.8

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

$\sqrt{1 + 1 + 4 + 9 + 0 + 1 + 16 + 9 + 1^{2} + 0^{2} + 3^{2} + 5^{2}}$

Step 2.9

One to any power is one.

$\sqrt{1 + 1 + 4 + 9 + 0 + 1 + 16 + 9 + 1 + 0^{2} + 3^{2} + 5^{2}}$

Step 2.10

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

$\sqrt{1 + 1 + 4 + 9 + 0 + 1 + 16 + 9 + 1 + 0 + 3^{2} + 5^{2}}$

Step 2.11

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

$\sqrt{1 + 1 + 4 + 9 + 0 + 1 + 16 + 9 + 1 + 0 + 9 + 5^{2}}$

Step 2.12

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

$\sqrt{1 + 1 + 4 + 9 + 0 + 1 + 16 + 9 + 1 + 0 + 9 + 25}$

Step 2.13

Add $1$ and $1$ .

$\sqrt{2 + 4 + 9 + 0 + 1 + 16 + 9 + 1 + 0 + 9 + 25}$

Step 2.14

Add $2$ and $4$ .

$\sqrt{6 + 9 + 0 + 1 + 16 + 9 + 1 + 0 + 9 + 25}$

Step 2.15

Add $6$ and $9$ .

$\sqrt{15 + 0 + 1 + 16 + 9 + 1 + 0 + 9 + 25}$

Step 2.16

Add $15$ and $0$ .

$\sqrt{15 + 1 + 16 + 9 + 1 + 0 + 9 + 25}$

Step 2.17

Add $15$ and $1$ .

$\sqrt{16 + 16 + 9 + 1 + 0 + 9 + 25}$

Step 2.18

Add $16$ and $16$ .

$\sqrt{32 + 9 + 1 + 0 + 9 + 25}$

Step 2.19

Add $32$ and $9$ .

$\sqrt{41 + 1 + 0 + 9 + 25}$

Step 2.20

Add $41$ and $1$ .

$\sqrt{42 + 0 + 9 + 25}$

Step 2.21

Add $42$ and $0$ .

$\sqrt{42 + 9 + 25}$

Step 2.22

Add $42$ and $9$ .

$\sqrt{51 + 25}$

Step 2.23

Add $51$ and $25$ .

$\sqrt{76}$

Step 2.24

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 2.24.1

Factor $4$ out of $76$ .

$\sqrt{4 (19)}$

Step 2.24.2

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

$\sqrt{2^{2} \cdot 19}$

Step 2.25

Pull terms out from under the radical.

$2 \sqrt{19}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{19}$

Decimal Form:

$8.71779788 \dots$