Matrix Space

Vector Interpretation

Our Interpretation of vectors is something like this,

Say we have a vector $\vec{v}\in\mathbb{R}^3$
then each element of vector $\vec{v}$ represent an unique axis (say $$x,y$$ and $$z$$ ).
these axis can be anything your salary, your SAT score, your running speed (anything which can be quantify in a number).
Assume a line passing thrown each element in that vector $\vec{v}$ .
$\vec{v}=\begin{bmatrix} \color{red}{\bullet}\\ \color{blue}{\bullet}\\ \color{green}{\bullet}\\ \end{bmatrix}$ .
interpret these (red $\bullet$ , blue $\bullet$ and green $\bullet$ ) dots as real line $\in\mathbb{R}^1$

Matrix Interpretation

We can use our interpretation of vectors for matrices,

Say we have a $m\times n$ matrix say $A\in\mathbb{R}^{m\times n}$
then each element of our matrix $$A$$ represent an unique axis (say $u,v,w,\cdots$ ).
These axis can be anything your salary, your SAT score, your average running speed, average temperature in past 20 days, (anything which can be quantify in a number).
Assume a line passing thrown each element in this matrix $$A$$ .
$A=\begin{bmatrix} \color{red}{\bullet} & \color{brown}{\bullet} \\ \color{green}{\bullet} & \color{blue}{\bullet} \\ \end{bmatrix}$
interpret these (red $\bullet$ , blue $\bullet$ , green $\bullet$ and brown $\bullet$ ) dots as real line $\in\mathbb{R}^1$

Vector interpretation of a Matrix

We can interpret matrices as vectors.

Say we have a $2\times 2$ matrix say $A\in\mathbb{R}^{2\times 2}$
$A=\begin{bmatrix} \color{red}{\bullet} & \color{brown}{\bullet} \\ \color{green}{\bullet} & \color{blue}{\bullet} \\ \end{bmatrix}$
We can interpret it as vector say $\vec{v}\in\mathbb{R}^4$ as,
$\vec{v}=\begin{bmatrix} \color{red}{\bullet}\\ \color{brown}{\bullet}\\ \color{green}{\bullet}\\ \color{blue}{\bullet}\\ \end{bmatrix}$ .

So we can see that our matrices are the vectors.

Matrix space

Think of all

3\times 3

matrices, there are not any limitation to elements, they can be anything, there are

\infty

of them.

These $3\times 3$ matrices can we written as vectors $\vec{v}\in\mathbb{R}^9$

We can add those matrices, and if we add them we got another

3\times 3

matrix.

We can multiply matrices by a scaler, and if we multiply a scaler to a

3\times 3

matrix we get another

3\times 3

matrix.

We can take there linear combinations and the resultant matrix will still be a

3\times 3

matrix

So the space of all $3\times 3$ matrices is a vector space

Let's denote this space of all

3\times 3

matrices as

\mathcal{M}

.
Now let's see sub spaces of

\mathcal{M}

Lower Triangular

All

3\times 3

Lower Triangular matrices

\subset

All

3\times 3

matrices.
Is the space of all

3\times 3

Lower Triangular matrices is a subspace of all

3\times 3

matrices?

We know that All $3\times 3$ Lower Triangular matrices $\subset$ All $3\times 3$ matrices, then,
Space of all Lower Triangular matrices is a subspace of all $3\times 3$ matrices if,
All Lower Triangular $3\times 3$ matrices form a vector space?

Do all Lower Triangular $3\times 3$ matrices form a vector space?.

We can add two

3\times 3

Lower Triangular matrices, we got another

3\times 3

Lower Triangular matrix.

$\begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} + \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix}$

We can multiply a

3\times 3

Lower Triangular matrices by a scaler, we get another

3\times 3

Lower Triangular matrix.

$C\cdot \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix}$

We can take there linear combinations and the resultant matrix will still be a

3\times 3

matrix

$\alpha\cdot \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} + \beta\cdot \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix}$

So collectively we can say that all

3\times 3

Lower Triangular matrices forms a vector space.
So

The space all $3\times 3$ Lower Triangular matrices is a subspace of all $3\times 3$ matrices .

Symmetric Matrix

All

3\times 3

Symmetric matrices

\subset

All

3\times 3

matrices.
Is the space of all

3\times 3

Symmetric matrices is a subspace of all

3\times 3

matrices?

We know that All $3\times 3$ Symmetric matrices $\subset$ All $3\times 3$ matrices, then,
Space of all Symmetric matrices is a subspace of all $3\times 3$ matrices if,
All Symmetric $3\times 3$ matrices form a vector space?

Do all Symmetric $3\times 3$ matrices form a vector space?.

We can add two

3\times 3

Symmetric matrices, we got another

3\times 3

Symmetric matrix.

$\begin{bmatrix} \mathbb{R} & \color{red}{\mathbb{R}} & \color{blue}{\mathbb{R}} \\ \color{red}{\mathbb{R}} & \mathbb{R} & \color{green}{\mathbb{R}} \\ \color{blue}{\mathbb{R}} & \color{green}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} + \begin{bmatrix} \mathbb{R} & \color{brown}{\mathbb{R}} & \color{magenta}{\mathbb{R}} \\ \color{brown}{\mathbb{R}} & \mathbb{R} & \color{orange}{\mathbb{R}} \\ \color{magenta}{\mathbb{R}} & \color{orange}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} 2\mathbb{R} & \color{red}{\mathbb{R}}+\color{brown}{\mathbb{R}} & \color{blue}{\mathbb{R}}+\color{magenta}{\mathbb{R}} \\ \color{red}{\mathbb{R}}+\color{brown}{\mathbb{R}} & 2\mathbb{R} & \color{green}{\mathbb{R}}+\color{orange}{\mathbb{R}} \\ \color{blue}{\mathbb{R}}+\color{magenta}{\mathbb{R}} & \color{green}{\mathbb{R}}+\color{orange}{\mathbb{R}} & 2\mathbb{R} \\ \end{bmatrix}$

We can multiply a

3\times 3

Symmetric matrices by a scaler, we get another

3\times 3

Symmetric matrix.

$C\cdot \begin{bmatrix} \mathbb{R} & \color{red}{\mathbb{R}} & \color{blue}{\mathbb{R}} \\ \color{red}{\mathbb{R}} & \mathbb{R} & \color{green}{\mathbb{R}} \\ \color{blue}{\mathbb{R}} & \color{green}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} C\cdot\mathbb{R} & \color{red}{C\cdot\mathbb{R}} & \color{blue}{C\cdot\mathbb{R}} \\ \color{red}{C\cdot\mathbb{R}} & C\cdot\mathbb{R} & \color{green}{C\cdot\mathbb{R}} \\ \color{blue}{C\cdot\mathbb{R}} & \color{green}{C\cdot\mathbb{R}} & C\cdot\mathbb{R} \\ \end{bmatrix}$

We can take there linear combinations and the resultant matrix will still be a

3\times 3

matrix

$\alpha\cdot \begin{bmatrix} \mathbb{R} & \color{red}{\mathbb{R}} & \color{blue}{\mathbb{R}} \\ \color{red}{\mathbb{R}} & \mathbb{R} & \color{green}{\mathbb{R}} \\ \color{blue}{\mathbb{R}} & \color{green}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} + \beta\cdot \begin{bmatrix} \mathbb{R} & \color{brown}{\mathbb{R}} & \color{magenta}{\mathbb{R}} \\ \color{brown}{\mathbb{R}} & \mathbb{R} & \color{orange}{\mathbb{R}} \\ \color{magenta}{\mathbb{R}} & \color{orange}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} = \begin{bmatrix} \alpha\cdot\mathbb{R} + \beta\cdot\mathbb{R} & \color{red}{\alpha\cdot\mathbb{R}}+\color{brown}{\beta\cdot\mathbb{R}} & \color{blue}{\alpha\cdot\mathbb{R}}+\color{magenta}{\beta\cdot\mathbb{R}} \\ \color{red}{\alpha\cdot\mathbb{R}}+\color{brown}{\beta\cdot\mathbb{R}} & \alpha\cdot\mathbb{R} + \beta\cdot\mathbb{R} & \color{green}{\alpha\cdot\mathbb{R}}+\color{orange}{\beta\cdot\mathbb{R}} \\ \color{blue}{\alpha\cdot\mathbb{R}}+\color{magenta}{\beta\cdot\mathbb{R}} & \color{green}{\alpha\cdot\mathbb{R}}+\color{orange}{\beta\cdot\mathbb{R}} & \alpha\cdot\mathbb{R} + \beta\cdot\mathbb{R} \\ \end{bmatrix}$

So collectively we can say that all

3\times 3

Symmetric matrices forms a vector space.
So

The space all $3\times 3$ Symmetric matrices is a subspace of all $3\times 3$ matrices .

Similarly The space all $3\times 3$ Diagonal matrices is a subspace of all $3\times 3$ matrices .