Let f be a fixed 3×2 matrix, and H be the set of matrices A belonging to a 2×4 matrix. If we assume that the property FA = O holds true, show that H is a subspace of M2×4. Here O represents a zero matrix of order 3×4.

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 $u$ , $v$ , $k u$ , $k v$ 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 $k u$ $\in$ $V$ where $k$ is scalar)

Expert Answer

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

$A = 0$

Then for any matrix F:

$F A = 0$ .

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:

$F (A_{1} + A_{2}) = F A_{1} + F A_{2} = 0 + 0 = 0$

Since:

$F A_{1} = 0, F A_{2} = 0 \in H$

and also:

$F A_{1} + F A_{2} = 0 \in H$

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:

$F (c A) = c (F A)$

Since:

$A \in H$

And:

$c (F A) = c (0) = 0 \in H$

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 $R^{2}$ .

Expert Answer

Numerical Result

Example

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