Explained: "Construct a matrix whose column space contains (1,..."

High School
Algebra
Pre-Algebra
Factors and multiples

Wribreeminsl

Answered question

2021-08-05

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

Answer & Explanation

lamusesamuset

Skilled2021-08-06Added 93 answers

Step 1
Consider the matrix $[\begin{matrix} 1 & 0 & 0 & - 1 \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 1 & - 1 \end{matrix}]$
Step 2
Here, $x_{4}$ is a free variable and $x_{1}, x_{2}, x_{3}$ are not. Column space of A contain all combination of vectors $[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]$ .
Therefore, it contains vector $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ .
Thus, we consider the system $[\begin{matrix} 1 & 0 & 0 & - 1 \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 1 & - 1 \end{matrix}] = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$
Step 3
$x_{1} - x_{4} = 0 \Rightarrow x_{1} = x_{4}$
$x_{2} - x_{4} = 0 \Rightarrow x_{2} = x_{4}$
$x_{3} - x_{4} = 0 \Rightarrow x_{3} = x_{4}$
$N (A) = x_{4} [\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}]$
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 $A = [\begin{matrix} 1 & 0 & 0 & - 1 \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 1 & - 1 \end{matrix}]$ .

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