Linear Algebra Examples

$[\begin{array}{c} 1 & 2 & 5 & 9 \\ 11 & 14 & 19 & 5 \\ 7 & 1 & 3 & 6 \\ 13 & 17 & 21 & 0 \end{array}]$

Step 1

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

$\sqrt{1^{2} + 2^{2} + 5^{2} + 9^{2} + 11^{2} + 14^{2} + 19^{2} + 5^{2} + 7^{2} + 1^{2} + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

One to any power is one.

$\sqrt{1 + 2^{2} + 5^{2} + 9^{2} + 11^{2} + 14^{2} + 19^{2} + 5^{2} + 7^{2} + 1^{2} + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.2

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

$\sqrt{1 + 4 + 5^{2} + 9^{2} + 11^{2} + 14^{2} + 19^{2} + 5^{2} + 7^{2} + 1^{2} + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.3

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

$\sqrt{1 + 4 + 25 + 9^{2} + 11^{2} + 14^{2} + 19^{2} + 5^{2} + 7^{2} + 1^{2} + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.4

Raise $9$ to the power of $2$ .

$\sqrt{1 + 4 + 25 + 81 + 11^{2} + 14^{2} + 19^{2} + 5^{2} + 7^{2} + 1^{2} + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.5

Raise $11$ to the power of $2$ .

$\sqrt{1 + 4 + 25 + 81 + 121 + 14^{2} + 19^{2} + 5^{2} + 7^{2} + 1^{2} + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.6

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

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 19^{2} + 5^{2} + 7^{2} + 1^{2} + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.7

Raise $19$ to the power of $2$ .

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 5^{2} + 7^{2} + 1^{2} + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.8

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

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 7^{2} + 1^{2} + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.9

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

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1^{2} + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.10

One to any power is one.

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 3^{2} + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.11

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

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 6^{2} + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.12

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

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 13^{2} + 17^{2} + 21^{2} + 0^{2}}$

Step 2.13

Raise $13$ to the power of $2$ .

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 17^{2} + 21^{2} + 0^{2}}$

Step 2.14

Raise $17$ to the power of $2$ .

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 21^{2} + 0^{2}}$

Step 2.15

Raise $21$ to the power of $2$ .

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 441 + 0^{2}}$

Step 2.16

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

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 441 + 0}$

Step 2.17

Add $1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 441$ and $0$ .

$\sqrt{1 + 4 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 441}$

Step 2.18

Add $1$ and $4$ .

$\sqrt{5 + 25 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 441}$

Step 2.19

Add $5$ and $25$ .

$\sqrt{30 + 81 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 441}$

Step 2.20

Add $30$ and $81$ .

$\sqrt{111 + 121 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 441}$

Step 2.21

Add $111$ and $121$ .

$\sqrt{232 + 196 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 441}$

Step 2.22

Add $232$ and $196$ .

$\sqrt{428 + 361 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 441}$

Step 2.23

Add $428$ and $361$ .

$\sqrt{789 + 25 + 49 + 1 + 9 + 36 + 169 + 289 + 441}$

Step 2.24

Add $789$ and $25$ .

$\sqrt{814 + 49 + 1 + 9 + 36 + 169 + 289 + 441}$

Step 2.25

Add $814$ and $49$ .

$\sqrt{863 + 1 + 9 + 36 + 169 + 289 + 441}$

Step 2.26

Add $863$ and $1$ .

$\sqrt{864 + 9 + 36 + 169 + 289 + 441}$

Step 2.27

Add $864$ and $9$ .

$\sqrt{873 + 36 + 169 + 289 + 441}$

Step 2.28

Add $873$ and $36$ .

$\sqrt{909 + 169 + 289 + 441}$

Step 2.29

Add $909$ and $169$ .

$\sqrt{1078 + 289 + 441}$

Step 2.30

Add $1078$ and $289$ .

$\sqrt{1367 + 441}$

Step 2.31

Add $1367$ and $441$ .

$\sqrt{1808}$

Step 2.32

Rewrite $1808$ as $4^{2} \cdot 113$ .

Tap for more steps...

Step 2.32.1

Factor $16$ out of $1808$ .

$\sqrt{16 (113)}$

Step 2.32.2

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} \cdot 113}$

Step 2.33

Pull terms out from under the radical.

$4 \sqrt{113}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{113}$

Decimal Form:

$42.52058325 \dots$