Question

1. suppose ( a ) is the matrix a = \begin{bmatrix} 0 & 1 & 2 & 2 \\ 0 & 3 & 8 & 7 \\ 0 & 0 & 4 & 2 end{bmatrix} . (a) (16 points) find all special solutions to ( ax = 0 ) and describe in words the whole nullspace of ( a ).

Explanation:

Step1: Row reduce matrix A

Perform row operations to get row-echelon form:

$$ A =
LATEXBLOCK0
\xrightarrow{R2=R2-3R1}
LATEXBLOCK1
\xrightarrow{R3=R3-2R2}
LATEXBLOCK2
\xrightarrow{R2=\frac{1}{2}R2}
LATEXBLOCK3
\xrightarrow{R1=R1-2R2}
LATEXBLOCK4
$$

Step2: Identify free/pivot variables

Pivot columns: 2, 3; pivot variables: $x_2, x_3$
Free columns: 1, 4; free variables: $x_1 = s$, $x_4 = t$ (where $s,t \in \mathbb{R}$)

Step3: Solve for pivot variables

From row-reduced form:
$x_2 + x_4 = 0 \implies x_2 = -t$
$x_3 + \frac{1}{2}x_4 = 0 \implies x_3 = -\frac{1}{2}t$

Step4: Write general solution

Express $\mathbf{x}$ as combination of free variables:

$$ \mathbf{x} =
LATEXBLOCK5
= s
LATEXBLOCK6
+ t
LATEXBLOCK7
$$

Special solutions are the vectors multiplied by $s$ and $t$.

Answer:

Special solutions:

$$\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}$$

and

$$\begin{bmatrix} 0 \\ -1 \\ -\frac{1}{2} \\ 1 \end{bmatrix}$$

The nullspace of $A$ is the set of all linear combinations of these two special solutions, forming a 2-dimensional subspace of $\mathbb{R}^4$.