Mark O. answered • 11/24/16
Tutor
5.0
(167)
Learn Physics, Math, and Comp Sci from Professional Scientist
Hi JW,
If X1 = [1 -1 1] and X2 = [1 0 -1], then the matrix A is
A = 1 -1 1
1 0 -1
Of course, this is the best way I can write A given the limitation of this editor.
To find the null space of A, you write the matrix equation
Ax = θ,
where θ represents the zero vector [0 0]T, where T stands for transpose, to make this a column vector.
To find x, which are all vectors in the null space of A, you augment A with θ and row reduce.
1 -1 1 | 0
1 0 -1 | 0
Multiply row 2 by -1.
1 -1 1 | 0
-1 0 1 | 0
-1 0 1 | 0
Hold the top row constant and add the top row to the second row.
1 -1 1 | 0
0 -1 2 | 0
0 -1 2 | 0
Multiply the second row by -1 again.
1 -1 1 | 0
0 1 -2 | 0
0 1 -2 | 0
Hold the second row constant and add it to the first row.
1 0 -1 | 0
0 1 -2 | 0
0 1 -2 | 0
This is as far as we can go. You want to make the left-most part of the matrix look like the identity matrix as closely as you can. Let us call the coordinate variables x1, x2 and x3, respectively. Then, these two equations read as
x1 - x3 = 0
x2 - 2x3 = 0
Or,
x1 = x3
x2 = 2x3
We are free to choose a value for independent variable x3. But, once we select a value for x3, then we determine the corresponding dependent values for x1 and x2. Then, the null space of A consists of vectors of the form:
[x3 2x3 x3]T, where the T means transpose to make them column vectors.
What does this mean physically? This means that any vector x that multiplies A to give the zero vector θ must be of the form [x3 2x3 x3]T, which is an infinite number of vectors since there is an infinite number of ways to select x3. Let's do a test. let x3 = 1. Then, the corresponding null space vector would be [1 2 1]T. A times this vector will yield the zero vector. Try this calculation for yourself to convince yourself what the null space means.
