Question

Problem 6. (10 points) Suppose that A is a matrix with seven columns and four rows. Suppose that the columns of A span R. Determine the number of vectors needed to form a basis for the nullspace of T. Justify your answer.

Answer 1

(when nothing is mentioned about T we can assume it is standard notation for linear transformation , from context it is clear that matrix of linear transformation is A)
Mat rise A has seven columns and four rowes. Order of Mathise A = So Matrix A can be viewed as Matrix of some linear Transformation T: R7 → R IRY T So rank Theoremi villity Rank. I t vel 7 rillity Rank T= Rank A and willity T = Nillity A So Rank A+ vullit A = 0 Rank A = Y column rank of A oo columns of A span IRY hence dimension of Column space for column rank of A is 4 or 2 So Rank A =

2 From equation 0 and villity A= 7 - 4 => [nrillity A = 3 Hence => [willity T= 3 (nullity to prillity ) Three vectors are needed to form a well space for it as well as