Let Matrix that is row equivalent to the matrix Find rank A

Let Matrix that is row equivalent to the matrix

Find rank A and dim Nul A

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

2. (10 points) Let matrix 1332-9 1- 2 -2 2-8 2 L3 4-1 11 -8 that is row equivalent to the matrix 1 0-3 5 01 0 1 2-1 0 B01 2 10 0 0 0 01 (a) (5 points) Find rank A and dim Nul A. (b) (5 points) Find a basis for the column space of A (Col A)

Solution

a.

rank A = dim row A

But A is row equivalent to B

dim row B=3 as last row is 0 and first three rows are clearly independent

Hence, rank A=3

by rank nullity theorem

rank A+nullity A=5

Hence, dim Nul A=5-3=2

b.

rank A=dim col A.=3

Hence basis for column space for A has 3 vectors.

From B we see first second and fifth columns are linearly independnet

Hence, in A also first ,second and fifth columns are linearly independent and they form the basis for Col A.