Question

Prove theorem 3 as follows: Given an $m\times n$ matrix A, an element in $\mathrm{Col}A$ has the form $Ax$ for some x in ${\mathbb{R}}^n$. Let $Ax$ and $A\mathrm{w}$ represent any two vectors in $\mathrm{Col}A$.

1. Explain why the zero vector is in $\mathrm{Col}A$.
2. Show that the vector $A\mathrm{x}+A\mathrm{w}$ is in $\mathrm{Col}A$.
3. Given a scalar $c$, show that $c\left(A\mathrm{x}\right)$ is in $\mathrm{Col}A$.

Short answer

a. The zero vector is in $\mathrm{Col}A$.

b. It is proved that the vector $A\mathrm{x}+A\mathrm{w}$ is in $\mathrm{Col}A$.

c. It is proved that $c\left(A\mathrm{x}\right)$ is in $\mathrm{Col}A$

Step-by-step solution

Explain that the zero vector is in $\mathrm{Col}A$

a)

It is given that in an $m\times n$ matrix, an element in $\mathrm{Col}A$ has the form $A\mathrm{x}$ for some x in ${\mathbb{R}}^n$.

Thezero vector is in $\mathrm{Col}A$ because of the equation $A0=0$.

Thus, the zero vector is in $\mathrm{Col}A$.

Show that the vector \(A{\mathop{\rm x}\nolimits}  + A{\mathop{\rm

b)

It is easy to find vectors in $\mathrm{Col}A$. Thecolumnsof A are displayed; others are formed from them.

The vector $A\mathrm{x}+A\mathrm{w}$ is in $\mathrm{Col}A$ because$A\mathrm{x}+A\mathrm{w}=A\left(\mathrm{x}+\mathrm{w}\right)$.

Thus, it is proved that the vector $A\mathrm{x}+A\mathrm{w}$ is in $\mathrm{Col}A$.

Show that $c\left(A\mathrm{x}\right)$ is in $\mathrm{Col}A$

c)

The vector $c\left(A\mathrm{x}\right)$ is in $\mathrm{Col}A$ because $c\left(A\mathrm{x}\right)=A\left(c\mathrm{x}\right)$.

Thus, it is proved that $c\left(A\mathrm{x}\right)$ is in $\mathrm{Col}A$.