# Vector Space

Consider a vector\(\vec{v}\in\mathbb{R}^d\)

, here \(\vec{v}\)

lives in \(d\)

dimensional space this space is a **vector space**.

For example for

\(d=3\)

, Vector shown in the blue is a vector

\(\vec{x}\in\mathbb{R}^3\)

. All the space surrounding this vector is \(3\)

-dimensional vector space.## Code to plot this vector (python)

```
import MultiVariable as mvar
import numpy as np
%matplotlib qt
m = mvar.MultiVariable()
vector = np.array([[1,1,1]])
m.plot_3D_vectors(vector)
## Adjusting axis limits
m.set_axes_limit((-2,2))
m.setZ_limit((-0.5,2))
```

Download MultiVariable classIf we multiply a vector

\(\vec{v}\)

by a constant say \(c\)

then the resulting vector is in direction of \(\vec{v}\)

(or in opposite direction).Say the resulting vector is

\(\vec{v}'\)

then we can say \(\vec{v}' = c \vec{v} \)

where \(c\in\mathbb{R}\)

, it is a line along vector \(\vec{v}\)

.Here you can see that

\(\vec{v}'\)

has it's own space inside that \(d\)

dimensional space, it's a vector space inside a vector space.Example for

\(d=3\)

Here you can see the vector

\(\vec{x}\in\mathbb{R}^3\)

has it's own vector space that is that line.## Code To plot this vector (python)

```
import MultiVariable as mvar
import numpy as np
%matplotlib qt
m = mvar.MultiVariable()
## Here line is defines as [x-start, x-end, y-start, y-end, z-start, z-end]
lines = np.array([[-2,2,-2,2,-2,2]])
m.plot_3D_lines(lines)
vector = np.array([[1,1,1]])
m.plot_3D_vectors(vector, plot_separately=False)
## Adjusting axis limits
m.set_axes_limit((-3,3))
```

Download MultiVariable classNow Consider a plane(passing through origin) inside a

\(3\)

-dimensional space, so that plane lives inside \(3\)

-dimensional vector space, but it has it's own vector space.## Code To plot this plane (python)

```
import MultiVariable as mvar
import numpy as np
%matplotlib qt
def f(x,y):
return -x -y
m = mvar.MultiVariable()
m.plot_surface_color_3D(f, plot_separately=True, alpha=0.7)
```

Download MultiVariable class\(x+y+z=0\)

has it's own vector space inside \(3\)

-dimensional vector space.If we add any two

\(d\)

-dimensional vectors, we get a \(d\)

-dimensional vector as a output.Say we have two vectors

\(\vec{v}\in\mathbb{R}^d\)

and \(\vec{w}\in\mathbb{R}^d\)

, then if we take there **linear**combination, then that

**linear**combination will have it's own vector space, but what that vector space looks like?

Well it depends, on how

\(\vec{v}\)

and \(\vec{w}\)

are oriented.**Case 1: if**

\(\vec{v}\)

is parallel to \(\vec{w}\)

If

\(\vec{v}\)

and \(\vec{w}\)

are parallel then \(\vec{w}= c \vec{v};\quad c\in\mathbb{R}\)

, So vector space is just a line.Example for

\(d=3\)

:Say the vectors are

\(\vec{v} = \begin{bmatrix} 1 \\ 1\\ 1\\ \end{bmatrix}\)

and \(\vec{w} = \begin{bmatrix} 2 \\ 2\\ 2\\ \end{bmatrix}\)

Then the vector space is just a line passing through these vectors.

## Code To plot this (python)

```
import MultiVariable as mvar
import numpy as np
%matplotlib qt
m = mvar.MultiVariable()
vectors = np.array([
[1,1,1],
[2,2,2]
])
origin = np.array([0,0,0])
m.plot_3D_vectors(vectors, origin, plot_separately=False)
# Structure of lines = [[x-start, x-end, y-start, y-end, z-start, z-end],...]
lines = np.array([[-2.5,2.5,-2.5,2.5,-2.5,2.5]])
m.plot_3D_lines(lines, plot_separately=False)
m.set_axes_limit((-3,3))
```

Download MultiVariable class**Case 2: if**

\(\vec{v}\)

is not parallel to \(\vec{w}\)

If

\(\vec{v}\)

and \(\vec{w}\)

are not parallel then the vector space of **linear**combination of

\(\vec{v}\)

and \(\vec{w}\)

is a plane.Example for

\(d=3\)

:Say the vectors are

\(\vec{v} = \begin{bmatrix} 2 \\ -1\\ -1\\ \end{bmatrix}\)

and \(\vec{w} = \begin{bmatrix} -1 \\ 2\\ -1\\ \end{bmatrix}\)

Then the vector space is the plane passing through these vectors.

## Code To plot this (python)

```
import MultiVariable as mvar
import numpy as np
%matplotlib qt
# X + Y + Z = 0
def f(x,y):
return -x -y
m = mvar.MultiVariable(count = 10, x_range=(-2,2), y_range=(-2,2), z_range=(-2,2))
vectors = np.array([
[2,-1,-1],
[-1,2,-1]
])
origin = np.array([0,0,0])
m.plot_3D_vectors(vectors, origin, plot_separately=False)
m.plot_surface_lines_3d(f, density = 100, plot_separately=False)
m.set_axes_limit((-3,3))
```

Download MultiVariable class## Vector Space Properties

So when can we say that a**space**can be a

**vector space**?

A space is a

**vector space**if it full fill these conditions:

- If there is a vector in that space and we multiply that vector with a constant \(c\in\mathbb{R}\)then the resulting vector must be in the space
- Say we took two vectors \(\vec{v}\)and\(\vec{w}\)then there sum must be in that space.

**not a vector space**.

Think of a 2 dimensional space where

\(x\gt0\)

and \(y\gt0\)

We can add vector safely and we don't go out of the space.

What about multiplying a constant to a vector, take a vector in that space

\(\vec{v}=\begin{bmatrix} 4 \\ 3\\ \end{bmatrix}\)

.if we multiply

\(\vec{v}\)

by \(-1\)

then if goes out of space.## Code To plot this (python)

```
import MultiVariable as mvar
import numpy as np
%matplotlib qt
m = mvar.MultiVariable()
x = np.array([0, 5])
y1 = np.array([5,5])
y2 = np.array([0, 0])
m.fill_between(x, y1, y2, alpha=0.3)
vectors = np.array([
[4,3],
[-4,-3],
])
origin = np.array([0,0])
m.plot_2D_vectors(vectors, origin, plot_separately=False, head_width=0.2, head_length=0.2)
m.set_axes_limit((-5,5))
```

Download MultiVariable classAVector space inside a vector space is referred as a subspace.vector spacemust pass through theorigin.

Possible subspace of a

\(2\)

-dimensional space.- All \(\mathbb{R}^2\)space
- All lines passing through origin.
- Origin itself(\(\vec{0}\))