The aim of this question is to comprehend the key **linear algebra** concepts of **vector spaces** and **vector subspaces**.

A **vector space** is defined as a **set of all vectors** that fulfill the **associative** and **commutative** properties for **vector addition** and **scalar multiplication** operations. The minimum no. of unique vectors required to describe a certain vector space is called **basis vectors**. A **vector space** is an n-dimensional space defined by **linear combinations** of basis vectors.

Mathematically, a vector space **V** must fulfill the following properties:

**– Commutative Property of Vector Addition:** $ u \ + \ v \ = \ v \ + \ u $ where $u$, $v$ are the vectors in $V$

**– Associative Property of Vector Addition:** $ ( \ u \ + \ v \ ) \ + \ w \ = \ u \ + \ ( \ v \ + \ w \ ) $ where $u$, $v$, $w$ are the vectors in $V$

**– Additive Identity:** $ u \ + \ 0 \ = \ 0 \ + \ u \ = \ u $ where $0$ is the additive identity of $V$

**– Additive Inverse:** $ u \ + \ v \ = \ v \ + \ u \ = 0 $ where $u$ and $v$ are the additive inverse of each other within $V$

**– Multiplicative Identity:** $ u \ \cdot \ 1 \ = \ 1 \ \cdot \ u \ = \ u $ where $1$ is the multiplicative identity of $V$

**– Distributive Property:** $ k \ \cdot \ ( \ u \ + \ v \ ) \ = \ k \ \cdot \ ( \ v \ + \ w \ ) \ = \ k \ \cdot \ u \ + \ k \ \cdot \ v $ where $k$ is a scalar multiple and $u$, $v$, $ku$, $kv$ are belong to $V$

A **subspace** $W$ is a subset of a vector space $V$ that **fulfills the following three properties**:

– $W$ must contain a **zero vector** (an element of $V$)

– $W$ must follow** closure property with respect to addition**. (i.e. if $u$, $v$ \in $V$ then $u \ + \ v$ $\in$ $V$)

– $W$ must follow **closure property with respect to scalar multiplication**. (i.e. if $u$ \in $V$ then $ku$ $\in$ $V$ where $k$ is scalar)

## Expert Answer

**Property (1):** Check if $H$ contains **zero vector.**

Let:

\[ A \ = \ 0 \]

Then for any matrix F:

\[ FA \ = \ 0 \].

So $H$ contains the zero vector.

**Property (1):** Check if $H$ is **closed w.r.t. vector addition**.

Let:

\[ A_1, \ A_2 \ \in \ H \]

Then, from distributive property of matrices:

\[ F(A_1 \ + \ A_2) \ = \ FA_1 \ + \ FA_2 \ = \ 0 \ + \ 0 \ = \ 0 \]

Since:

\[ FA_1 \ = \ 0, \ FA_2 \ = \ 0 \ \in \ H \]

and also:

\[ FA_1 \ + \ FA_2 \ = \ 0 \ \in \ H \]

So H is closed under addition.

**Property (3):** Check if $H$ is **closed w.r.t. scalar multiplication**.

Let:

\[ c \ \in \ R, \ A \ \in \ H \]

From scalar properties of matrices:

\[ F(cA) \ = \ c(FA) \]

Since:

\[ A \ \in \ H \]

And:

\[ c(FA) \ = \ c(0) \ = \ 0 \ \in \ H \]

So, $H$ is closed under scalar multiplication.

## Numerical Result

$H$ is a subspace of $M_{2 \times 4}$.

## Example

– Any plane $\in$ $R^2$ passing through the origin $(0, \ 0, \ 0)$ $\in$ $R^3$ is a subspace of $R^3$.

– Any line $\in$ $R^1$ passing through the origin $(0, \ 0, \ 0)$ $\in$ $R^3$ or $(0, \ 0)$ $\in$ $R^2$ is a subspace of both $R^3$ and $R^2$.