A is an m x n matrix Mark each statement True or False Justi

A is an m x n matrix. Mark each statement True or False. Justify each answer.

18. a. If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A. b. Row operations preserve the linear dependence relations among the rows of A. C. The dimension of the null space of A is the number of columns of A that are not pivot columns. d. The row space of A^T is the same as the column space of A.

Solution

18)

a) The first statement is FALSE, it will not be equal to basis for the column space of A, it will be the corresponding columns in A

b) The second statement is FALSE,

assuming matrix as A = [1 1; 1 1]

Let us say R2 -> R2-R1

Modified A = [1 1;0 0]

Earlier rows of A were same but now they are different

c) The third statment is TRUE

since the dimnesion of null space will b equal to number of free variables satisfying the equations Ax=0

which equals number of non-pivotted columns in matrix A

d) The fourth statement is TRUE

since when we take A^(T), the rows of A matrix becomes column of modified matrix, hence the row space of A^(T) will be equal to column space of matrix A