Matrix Subspaces

Question 1

How do you find the column space for matrix A?

Answer 1

Break down A into L&U, the column space is then the first r columns of L, where r is the number of non-zero rows in U (and is also the rank of the matrix)

Question 2

How do you find the row space of matrix A?

Answer 2

Break A down into L&U, and take all non-zero rows of U

Question 3

How do you find the null space of A?

Answer 3

The null space of A is every independent vector that solves Ax = 0, or LUx = 0
Since L is always invertible, this is the same as the solutions of Ux=0

Question 4

How do you find the left null space of A?

Answer 4

The left null space of A is the solution to y^tA=0
This is the same as L^ty = 0