Given lin independent columns of a matrix and asked to find

Given lin independent columns of a matrix and asked to find nullspace dimension and reduced echelon form of A

Let A be a 6 Times 5 matrix with linearly independent columns a_1, a_2 a_3 whose remaining columns satisfy a_4 = a_1 + 3a_2 + a_3 and a_5 = 2a_1 - a_3 What is the dimension of N(A)? Explain. Determine the reduced echelon form of A.

Solution

we will suppose a matrix

From here, above supposed to put it in equations, we will get this

x1 – 2x2 + 19x4 – 6x5 – 37x7 = 0 x3 – 6x4 + 2x5 + 6x7 = 0 x6 + 3x7 = 0

x1 = 2x2 – 19x4 + 6x5 + 37x7 x3 = 6x4 – 2x5 – 6x7 x6 = -3x7

a) Dimension of the column space = number of linearly independent columns = column rank = row rank = number of linearly independent rows = rank.