Operations on V_n

==2024-11-18

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Considerations for operations

We can perform various Mathematical Operations on $V_n$ .

1. Set for n-space
2. $\bar{x}$ and $\bar{y}$ are two n-tuples taken from $V_n$. We have, $\bar{x}=(x_1,x_2,x_3,\dots ,x_n)$ and ~ly, $\bar{y}$.

Theory

1. Addition
2. Scalar Multiplication
3. Subtraction
4. Multiplication

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Addition

$\bar{x}+\bar{y}=(x_1+y_1,x_2+y_2,x_3+y_3,\dots ,x_n+y_n)$ Here, Addition is well defined Q&A.

Additive Identity

If we add $\bar{0}$ to $\bar{x}$ we get $\bar{x}$ itself. So, $\bar{0}$ is an additive identity in $V_n$.

$\bar{0}$

n-tuple consisting of all elements $0$ can be interpreted in many ways. Some ways being:

1. Additive Identity
2. n-space Origin
3. Number
4. n-tuple

Scalar Multiplication

let $\alpha$ be a scalar, Then the scalar multiplication of $\bar{x}$ by $\alpha$ is:

$$\alpha \bar{x}=(\alpha x_1,\alpha x_2,\alpha x_3,\dots ,\alpha x_n)$$

Here, Scalar Multiplication is well defined Q&A.

Additive Inverse

When $\alpha =-1$ then we get $-\bar{x}$. When we add $-\bar{x}$ to $\bar{x}$ we get a n-tuple consisting of all zeros. $\bar{0}=(0,0,0,\dots ,0)$ here are n $0_s$.

Subtraction

There is no subtraction operation. But it is the combination of scalar multiplication of $\bar{y}$ with -1 and addition with $\bar{x}$ like so: $\bar{x}-\bar{y}=\bar{x}+(-1)\bar{y}$

Multiplication

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Examples

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Implementation

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Q&A

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