Question

1. suppose a is the matrix

$$a = \
LATEXBLOCK0
.$$

(a) (16 points) find all special solutions to $ax = 0$ and describe in words the whole nullspace of $a$.

Explanation:

Step1: Identify pivot columns

Pivot columns are 2nd,3rd,4th (non-zero rows' leading 1s/non-zeros).

Step2: Set free variable $x_1=t$

Free variable: $x_1$ (column 1 has no pivot). Let $x_1=t$.

Step3: Solve for pivot variables

From row3: $4x_3 + 2x_4 = 0 \Rightarrow 2x_3 + x_4 = 0$
From row2: $3x_2 + 8x_3 +7x_4=0$. Substitute $x_4=-2x_3$: $3x_2 +8x_3 -14x_3=0 \Rightarrow 3x_2-6x_3=0 \Rightarrow x_2=2x_3$
From row1: $x_2 +2x_3 +2x_4=0$. Substitute $x_2=2x_3,x_4=-2x_3$: $2x_3+2x_3-4x_3=0$ (consistent).

Step4: Choose $x_3=0$ for special solution

Set $t=1,x_3=0$: $x_2=0,x_4=0$. Special solution:

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

.

Step5: Describe nullspace

Nullspace: All $t

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

$ (1D line through origin).

Answer:

Special solution:

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

; Nullspace is the set of all scalar multiples of

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

.