Final answer:
The explicit description of the null space of matrix a is spanned by the vectors [4t, 3t, t], where t is a scalar.
Step-by-step explanation:
To find an explicit description of the null space of matrix a, we need to solve the equation Ax = 0, where A is the matrix representation of a. The null space of a matrix consists of all vectors x that satisfy this equation. Let's solve the equation:
First, we write the augmented matrix [A | 0].
2 1 -1 | 0
-4 -2 2 | 0
Next, let's perform row operations to bring the matrix to row-echelon form.
Divide Row 1 by 2: 1/2 * R1 -> R1
R1: 1 1/2 -1/2 | 0
-4 -2 2 | 0
Add Row 1 to Row 2: R2 + R1 -> R2
R1: 1 1/2 -1/2 | 0
0 -1/2 3/2 | 0
Multiply Row 2 by -2: -2 * R2 -> R2
1 1/2 -1/2 | 0
0 1 -3 | 0
Add Row 2 to Row 1: R1 + R2 -> R1
1 2 -4 | 0
0 1 -3 | 0
Now we can see that the second column is a leading 1 column, so the free variable is x3. We can express x1 and x2 in terms of x3.
x3 is a free variable, let x3 = t.
x2 = 3t
x1 = 4t
Therefore, the null space of matrix a is spanned by the vectors [4t, 3t, t], where t is a scalar.