Question

# Construct a matrix whose column space contains (1, 1, 1) and whose nullspace is the line of multiples of (1, 1, 1, 1).

Factors and multiples
Construct a matrix whose column space contains (1, 1, 1) and whose nullspace is the line of multiples of (1, 1, 1, 1).

2021-08-06

Step 1
Consider the matrix $$\begin{bmatrix}1 & 0 & 0 & -1 \\0 & 1 & 0 & -1\\ 0 & 0 & 1 & -1 \end{bmatrix}$$
Step 2
Here, $$\displaystyle{x}_{{{4}}}$$ is a free variable and $$\displaystyle{x}_{{{1}}},{x}_{{{2}}},{x}_{{{3}}}$$ are not. Column space of A contain all combination of vectors $$\begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 1 \end{bmatrix}$$.
Therefore, it contains vector $$\begin{bmatrix}1 \\ 1 \\ 1 \end{bmatrix}$$.
Thus, we consider the system $$\begin{bmatrix}1 & 0 & 0 & -1 \\0 & 1 & 0 & -1\\ 0 & 0 & 1 & -1 \end{bmatrix}=\begin{bmatrix}x_{1} \\x_{2}\\ x_{3} \end{bmatrix}=\begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}$$
Step 3
$$\displaystyle{x}_{{{1}}}-{x}_{{{4}}}={0}\Rightarrow{x}_{{{1}}}={x}_{{{4}}}$$
$$\displaystyle{x}_{{{2}}}-{x}_{{{4}}}={0}\Rightarrow{x}_{{{2}}}={x}_{{{4}}}$$
$$\displaystyle{x}_{{{3}}}-{x}_{{{4}}}={0}\Rightarrow{x}_{{{3}}}={x}_{{{4}}}$$
$$N(A)=x_{4} \begin{bmatrix}1\\1\\1\\1 \end{bmatrix}$$
So null space of A consists of all multiples of (1,1,1,1).
A matrix whose column space matrix contains (1,1,1) and whose null space is the line of multiples of (1,1,1,1) is $$A=\begin{bmatrix}1 & 0 & 0 & -1 \\0 & 1 & 0 & -1\\0 & 0 & 1 & -1 \end{bmatrix}$$.