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

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

a)

It is given that in an \(m \times n\) matrix, an element in \({\mathop{\rm Col}\nolimits} A\) has the form \(A{\mathop{\rm x}\nolimits} \) for some x in \({\mathbb{R}^n}\).

Thezero vector is in \({\mathop{\rm Col}\nolimits} A\) because of the equation \(A0 = 0\).

Thus, the zero vector is in \({\mathop{\rm Col}\nolimits} A\).

b)

It is easy to find vectors in \({\mathop{\rm Col}\nolimits} A\). Thecolumnsof A are displayed; others are formed from them.

The vector \(A{\mathop{\rm x}\nolimits} + A{\mathop{\rm w}\nolimits} \) is in \({\mathop{\rm Col}\nolimits} A\) because\(A{\mathop{\rm x}\nolimits} + A{\mathop{\rm w}\nolimits} = A\left( {{\mathop{\rm x}\nolimits} + {\mathop{\rm w}\nolimits} } \right)\).

Thus, it is proved that the vector \(A{\mathop{\rm x}\nolimits} + A{\mathop{\rm w}\nolimits} \) is in \({\mathop{\rm Col}\nolimits} A\).

c)

The vector \(c\left( {A{\mathop{\rm x}\nolimits} } \right)\) is in \({\mathop{\rm Col}\nolimits} A\) because \(c\left( {A{\mathop{\rm x}\nolimits} } \right) = A\left( {c{\mathop{\rm x}\nolimits} } \right)\).

Thus, it is proved that \(c\left( {A{\mathop{\rm x}\nolimits} } \right)\) is in \({\mathop{\rm Col}\nolimits} A\).