Linear Algebra Examples

[\begin{matrix} 2 i - 3 & 0 & 2 \end{matrix}]

$[\begin{array}{c} 2 i - 3 & 0 & 2 \end{array}]$ ,

[\begin{matrix} 0 & 1 & 2 - i \end{matrix}]

$[\begin{array}{c} 0 & 1 & 2 - i \end{array}]$

Step 1

The distance between two vectors

u⃗

$u⃗$ and

v⃗

$v⃗$ in

C^{n}

$ℂ^{n}$ is defined to be

| | u⃗ - v⃗ | |

$| | u⃗ - v⃗ | |$ which is the Euclidean norm of the difference

u⃗ - v⃗

$u⃗ - v⃗$ .

d (u⃗, v⃗) = | | u⃗ - v⃗ | | = \sqrt{{| {u⃗}_{1} - {v⃗}_{1} |}^{2} + {| {u⃗}_{2} - {v⃗}_{2} |}^{2} + \dots + {| {u⃗}_{n} - {v⃗}_{n} |}^{2}}

$d (u⃗, v⃗) = | | u⃗ - v⃗ | | = \sqrt{{| {u⃗}_{1} - {v⃗}_{1} |}^{2} + {| {u⃗}_{2} - {v⃗}_{2} |}^{2} + \dots + {| {u⃗}_{n} - {v⃗}_{n} |}^{2}}$

Step 2

Find the norm of the difference

u⃗ - v⃗

$u⃗ - v⃗$ where

u⃗ = [\begin{matrix} 2 i - 3 & 0 & 2 \end{matrix}]

$u⃗ = [\begin{array}{c} 2 i - 3 & 0 & 2 \end{array}]$ and

v⃗ = [\begin{matrix} 0 & 1 & 2 - i \end{matrix}]

$v⃗ = [\begin{array}{c} 0 & 1 & 2 - i \end{array}]$ .

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Step 2.1

Create a vector of the difference.

⎡ ⎢ ⎣ \begin{matrix} 2 i - 3 - 0 0 - 1 2 - (2 - i) \end{matrix} ⎤ ⎥ ⎦

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

Step 2.2

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

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

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

Step 2.3

Simplify.

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Step 2.3.1

Subtract

0

$0$ from

2 i - 3

$2 i - 3$ .

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

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

Step 2.3.2

Rearrange terms.

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

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

Step 2.3.3

Use the formula

| a + b i | = \sqrt{a^{2} + b^{2}}

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

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

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

Step 2.3.4

Raise

- 3

$- 3$ to the power of

2

$2$ .

\sqrt{{\sqrt{9 + 2^{2}}}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}

$\sqrt{{\sqrt{9 + 2^{2}}}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.5

Raise

2

$2$ to the power of

2

$2$ .

\sqrt{{\sqrt{9 + 4}}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}

$\sqrt{{\sqrt{9 + 4}}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.6

Add

9

$9$ and

4

$4$ .

\sqrt{{\sqrt{13}}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}

$\sqrt{{\sqrt{13}}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7

Rewrite

{\sqrt{13}}^{2}

${\sqrt{13}}^{2}$ as

13

$13$ .

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Step 2.3.7.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{13}

$\sqrt{13}$ as

13^{\frac{1}{2}}

$13^{\frac{1}{2}}$ .

\sqrt{{(13^{\frac{1}{2}})}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}

$\sqrt{{(13^{\frac{1}{2}})}^{2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\sqrt{13^{\frac{1}{2} \cdot 2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}

$\sqrt{13^{\frac{1}{2} \cdot 2} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{13^{\frac{2}{2}} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}

$\sqrt{13^{\frac{2}{2}} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7.4

Cancel the common factor of

2

$2$ .

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Step 2.3.7.4.1

Cancel the common factor.

$\sqrt{13^{\frac{2}{2}} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7.4.2

Rewrite the expression.

$\sqrt{13^{1} + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.7.5

Evaluate the exponent.

$\sqrt{13 + {(0 - 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.8

Subtract

$1$ from

$0$ .

$\sqrt{13 + {(- 1)}^{2} + {| 2 - (2 - i) |}^{2}}$

Step 2.3.9

Raise

$- 1$ to the power of

$2$ .

$\sqrt{13 + 1 + {| 2 - (2 - i) |}^{2}}$

Step 2.3.10

Simplify each term.

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Step 2.3.10.1

Apply the distributive property.

$\sqrt{13 + 1 + {| 2 - 1 \cdot 2 - - i |}^{2}}$

Step 2.3.10.2

Multiply

$- 1$ by

$2$ .

$\sqrt{13 + 1 + {| 2 - 2 - - i |}^{2}}$

Step 2.3.10.3

Multiply

$- 1$ by

$- 1$ .

$\sqrt{13 + 1 + {| 2 - 2 + 1 i |}^{2}}$

Step 2.3.10.4

Multiply

$i$ by

$1$ .

$\sqrt{13 + 1 + {| 2 - 2 + i |}^{2}}$

Step 2.3.11

Subtract

$2$ from

$2$ .

$\sqrt{13 + 1 + {| 0 + i |}^{2}}$

Step 2.3.12

Add

$0$ and

$i$ .

$\sqrt{13 + 1 + {| i |}^{2}}$

Step 2.3.13

Use the formula

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

$\sqrt{13 + 1 + {\sqrt{0^{2} + 1^{2}}}^{2}}$

Step 2.3.14

Raising

$0$ to any positive power yields

$0$ .

$\sqrt{13 + 1 + {\sqrt{0 + 1^{2}}}^{2}}$

Step 2.3.15

One to any power is one.

$\sqrt{13 + 1 + {\sqrt{0 + 1}}^{2}}$

Step 2.3.16

Add

$0$ and

$1$ .

$\sqrt{13 + 1 + {\sqrt{1}}^{2}}$

Step 2.3.17

Any root of

$1$ is

$1$ .

$\sqrt{13 + 1 + 1^{2}}$

Step 2.3.18

One to any power is one.

$\sqrt{13 + 1 + 1}$

Step 2.3.19

Add

$13$ and

$1$ .

$\sqrt{14 + 1}$

Step 2.3.20

Add

$14$ and

$1$ .

$\sqrt{15}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{15}$

Decimal Form:

$3.87298334 \dots$

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