# Range & Null space of A matrix

1. Aug 4, 2011

### squenshl

1. The problem statement, all variables and given/known data
Let x $\in$ RN, y $\in$ RM & A $\in$ RMxN be a matrix. Denote the columns of A by Ak, k = 1,...,N. Let R(A) & N(A) be the range & null space of A respectively.
a) How do the colmuns of A relate to the range of A?
b) Your task is to find the solution to the problem y = Ax, where y & A are known & M = N. What role do R(A) & N(A) play?
c) Let RN $\ni$ x = (x1,x2), x1 $\in$ RN1, x2 $\in$ RN2 & N1 + N2 = N. Let A $\in$ RN1xN & consider the problem Ax = 0. Assume that you know x2. Solve for x

2. Relevant equations

3. The attempt at a solution
a) This is easy, the column space of A is just the range of A.
b) Do we just use the definitions of R(A) and N(A)?
c) I have no idea on this one.

2. Aug 5, 2011

### lanedance

be careful mixing N(A)=Nullspace and N=number of columns - I would use n instead for the number of columns, m for rows

3. Aug 5, 2011

### lanedance

for b), what would happen if y is not in the range of A?

4. Aug 5, 2011

### lanedance

for c) consider writing a as 4 matrices to understand how it works
$$\begin{pmatrix} B & C \\ D & E \end{pmatrix}$$

with
B - n1 x n1
C - n1 x n2
D - n2 x n1
E - n2 x n2

then consider the product
$$Ax = \begin{pmatrix} B & C \\ D & E \end{pmatrix}x = \begin{pmatrix} B & C \\ D & E \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$

bit of an abuse of notation, but hopefully its clear what we're trying to do

5. Aug 6, 2011

### squenshl

Bx1 + Cx2 = 0
Dx1 + Ex2 = 0

Is that correct mate, if so I'm lost on what to do next?

6. Aug 14, 2011

### squenshl

Still lost on this question.

7. Aug 15, 2011

### squenshl

If x2 is known what does that mean for x1 and in turn x.

8. Aug 22, 2011

### lanedance

well solving for x is essentially solving for x_1 as x_2 is known

9. Aug 22, 2011

### lanedance

Now re-arranging the equation we get

$$Ax = \begin{pmatrix} B & C \\ D & E \end{pmatrix}x = \begin{pmatrix} B & C \\ D & E \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = 0$$

$$Ax = \begin{pmatrix} B \\ D \end{pmatrix}x_1 = - \begin{pmatrix} C \\ E \end{pmatrix} x_2$$

this is a system of N equations with N1 unknowns