Let A be a 5 times 7 matrix What is the maximal possible ran

Let A be a 5 times 7 matrix. What is the maximal possible rank of A? If the reduced row echelon form of A has three nonzero rows, what is the dimension of the null space of A? If the matrix transformation x rightarrow Ax is onto, what is the dimension of the null space of A? If the null space of A is 3-dimensional, does the equation Ax = b have a solution for every b element of R^5? Why or why not? True or false, with explanation:

Solution

a) The minimum of 5,7 is 5 so the maximum possible rank of A is 5

b) As there are three nonzero rows so there will be two zero rows..

Hence null space has dimension 2

c) For this transformation to be onto we must have null space of dimension 0.

d) No as the rank of A and rank of the augmented matrix may not be same