I am stuck on this question, I guess my brain only mastered one way matrix manipulation. I would love to get to the bottom of the things here, appreciate your help! This is an exam practice question, and I don't have a solution to it.

Question.

A system of linear equations Ax=d is known to have the following solution:

$\displaystyle x=(1,2,0,-1,0)^T+s(2,1,1,0,0)^T+t(1,1,0,-1,1)^T$.

Assume that A is an m x n matrix. Let $\displaystyle c_1, c_2, ... c_n$ denote the columns of A.

Answer the following questions, or, if there is insufficient information to answer the question, say so.

(1) What number is n?

(2) What number is m?

(3) What (number) is the rank of A?

(4) Write down a basis for the nullspace of A, N(A).

(5) Which columns $\displaystyle c_i$ form a basis of the range, R(A)?

(6) Write down an expression for d as a linear combinations of columns $\displaystyle c_i$.

(7) Write down a non-trivial linear combination of columns $\displaystyle c_i$ which is equal to the zero vector.

My thoughts:

Ax=d means A(mx5 matrix) x x(5x1 vector) = d(mx1 vector)

Further, if I put the given information into equation format, I get:

$\displaystyle x_1=1+2s+t$

$\displaystyle x_2=2+s+t$

$\displaystyle x_3=s$

$\displaystyle x_4=-1-t$

$\displaystyle x_5=t$

I have two free variables - x_3 and x_5 (s and t respectively) - therefore there are two zero rows in A.

Also, the columns 1, 2 and 4 are columns containing pivot variables $\displaystyle x_1, x_2,x_4$. Does it mean that there are 3 rows containing pivot variables also???

Finally, is there a way to write down A from the given solution? If I could do that, I'd answer all the questions below in an instant...

So far:

(1) n=5

(2) ??? 5 or 3, and I am not sure how to prove either

(3) ???

(4) columns containing free variables would form the basis for nullspace: $\displaystyle c_3, c_5$

(5) columns containing pivot variables would the basis for the range, ie $\displaystyle c_1, c_2, c_4$

(6) d as linear combination of the columns [tex]c_i[\math] would be $\displaystyle x_1c_1+x_2c_2+...+x_5c_5$ except that I probably need to insert the values for c's (((

(7) to find c's for Ax=0, I think I would need to solve Ax=0 for x (when I know A).