TRUE or FALSE a If A has more columns than rows then its col

TRUE or FALSE

a) If A has more columns than rows, then its columns cannot form a basis for col(A).

b)If A has more rows than columns, then its columns must form a basis for col(A).

c)If A and B are square matrices and rank(A) = rank(B), then rank(A2) = rank(B2).

d)If the row space of A and the null space of A are both lines, then A has two columns.

e) The column space of a matrix A is the set of b for which Ax = b is consistent.

Solution

c)True

Given that A and B are square matrics and A and B are similar rank

since A and B are similar there is a P such that A=PBP^-1

        by squaring on both sides we get

                                                      A²=(PBP^-1)²

                                                                           ⁼(PBP^-1)(PBP^-1)

                                                       =(PBP^-1PBP^-1)

                                                       =(PBBP^-1)

                                                      =(PB²P^-1)

so A²=B²

e)true

A is a system of linear equations Ax=b is a consistent then b is the set for column space of A

consider Ax=b where x=[x₁

                                    x₂

                                    \'

                                    \'

                                   x_n]

A=[c₁,c₂,c_3,-----c_n] where c₁,c₂,c_3,-----c_n denotes the columns of A

then we can write Ax=x₁c₁+x₂c₂+------+x_nc_n=b

so if Ax=b is consistent this means that there are x₁,x₂,---x_n satisfying the above equation

this means that b is the columa space of A

on the otherhand if b is the column space of A then it means we can write

b=x₁c₁+x₂c₂+------+x_nc_n

which means Ax=b has a solution.