Consider the matrix

Question

A =

2 6 3 1

2 1 0 −2

3 −2 1 1

0 6 2 0

(a) Find a basis for the row space of A.

(b) Find a basis for the column space of A.

Answer 1

I won't ruin the fun by giving the answer away since this is an older question, but to find the row space, you will need to find the reduced-row echelon form of the matrix first. Then, use the columns that fit the basis as your column space.

To find the reduced-row echelon form, you must complete these commands.

I usually write my commands next to the row, but I will only give the command I used, so I will write them with an equal sign going to the row for each step.

1. (1/2)(R1) = R1
2. R2 - 2(R1) = R2
3. R3 - 3R1 = R3
4. (-1/5)(R2) = R2
5. R1 - 3R2= R1
6. R3 + 11R2 = R3
7. R4 - 6R2 = R4
8. (10/31)(R3) = R3
9. R1 + (3/10)(R3) =R3
10. R2 - (3/5)(R3)
11. R4 + (8/5)(R3) = R4
12. (-31/14)(R4) = R4
13. R1 + (22/31)(R4) = R1
14. R2 + (18/31)(R4) = R2
15. R3 - (61/31)(R4) = R3

This should give you the identity matrix, and that will give you the solution.