In parts (a) to (d) below, mark the statement True or False. a. The columns of a matrix A are linearly independent if the equation Ax0 has the trivial solution. Choose the correct answer below. A. False. The columns of a matrix A are linearly independent only if the matrix equation Ax0 has some solution other than the trivial solution. B. True. If the columns are linearly independent, then Ax0 has the trivial solution. C. False. For every matrix A, Ax0 has the trivial solution. The columns of A are independent only if the equation has no solution other than the trivial solution. D. True. If a matrix equation has the trivial solution then there do not exist nonzero weights for the columns of A such thatb. If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S. A. False. If Sis linearly dependent then there is at least one vector that is not a linear combination of the other vectors, but the others may be linear combinations of each other. B. False. If an indexed set of vectors, S, is linearly dependent, then it is only necessary that one of the vectors is a linear combination of the other vectors in the set. C. True. If an indexed set of vectors, S, is linearly dependent, then at least one of the vectors can be written as a linear combination of other vectors in the set. Using the basic properties of equality, each of the vectors in the linear combination can also be written as a linear combination of those vectors. D. True. If Sis linearly dependent then for each j, vi, a vector in S, is a linear combination of the preceding vectors in S. c. The columns of any 4x 5 matrix are linearly dependent. A. True. 4x5 matrix has more columns than rows, and if a set contains more vectors than there are entries in each vector, then the set is linearly dependent. B. True. When a 4x5 matrix is written in reduced echelon form, there will be at least one row of zeros, so the columns of the matrix are linearly dependent. C. False. If a matrix has more rows than columns then the columns of the matrix are linearly dependent. D. False. If A is a 4x5 matrix then the matrix equation Ax = 0 is inconsistent because the reduced echelon augmented matrix has a row with all zeros except in the last column. d. If x and y are linearly independent, and if(x, y, z) is linearly dependent, then z is in Span{x, y}.A. False. If x and y are linearly independent, and (x, y, z) is linearly dependent, then z must be the zero vector. So z cannot be in Span{x, y} B. True. If {x, y, z) is linearly dependent, then z must be a linear combination of x and y because x and y are linearly independent. So z is in Span{x, y}. C. True. If(x, y, z) is linearly dependent and x and y are linearly independent, then z must be the zero vector. So z is in Span{x, y) D. False. Vector z cannot be in Span{x, y} because x and y are linearly independent