1. Determine whether each statement is TRUE or FALSE.(a)If A is a 5 x 4 matrix, then the linear system Ar = b represents a set of 5 equations with4 unknowns.(b)If the reduced echelon form of the augmented matrix [A b] is a 5 x 5 identity matrix, thenthe linear system Ar = b has a unique solution.(c)A linear system Ar b will always have at least one least squares solution.(eIf the collection of vectors is linearly independent, then it can not contain the zero vector.(f)If the non-homogeneous equation Ax = b has a unique solution, then the homogeneousequation Ar = 0 will only have the trivial solution.(g)If the matrix A has a pivot in every column, then the matrix transformation T(r) = Axmust be one-to-one.(h)Every collection of four vectors in a three-dimensional vector space is linearly dependent.(i)If a linear transformation T is onto then its range is equal to its co-domain.(i)be invertible.If A is a square matrix and the linear system Ar b has no free variables, then A must(k)If a vector space V contains a set of d linear independent vectors, then the dimension ofV can not be less than d.(1)The null space of a matrix A is always the same as the orthogonal complement of thecolumn space of A'.

Note: Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts for you. To get the remaining sub-part solved please repost the complete question and mention the sub-parts to be solved. Introduction: If A is a m × n matrix and Ax = b has exactly one solution, rank(A) equals n or rank(A)=n.We have to verify the true and false statements. (a) The statement is "If A is a 5 x 4 matrix, then the linear system Ax = b represnts a set of 5 equations with 4 unknowns. Let A be the matrix of order 5 x 4, i.e. A=abcda'b'c'd'a''b''c''d''efghe'f'g'h' and let x be the vector X=xyzw. Multiply A to X. AX=abcda'b'c'd'a''b''c''d''efghe'f'g'h'xyzw Which implies that, ax+by+cz+dw=ba'x+b'y+c'z+d'w=b'ax"+by"+cz"+dw"=b"ex+fy+gz+hw=b"'e'x+f'y+g'z+h'w=b"" Which shows that matrix A of order 5 x 4 with the system Ax = b represents the set of 5 equations and four unkowns. Hence the given statement is true.(b) The statement is "If the reduced echelon form of the augmented matrix A b is a 5 x 5 identity matrix, thenthe linear system Ax = b has a unique". Yes, this statement is true because the solution of system Ax = b with an identity matrix always has a unique solution.(c) The statement is "A linear system Ax = b will always have at least one least square solution". This statement is false, because if matrix has its rank equal to its number of columns, then it has no solution or only one solution.