Mark each statement True or False. Justify

Chapter , Problem 1E

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Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case.) In parts are vectors in a nonzero finite-dimensional vector space V, and a. The set of all linear combinations of is a vector spaceb. spans V , then S spans V .c. is linearly independent, then so is S.d. If S is linearly independent, then S is a basis for V.e. If Span S = V , then some subset of S is a basis for V .f. If dim V = p and Span S = V , then S cannot be linearly dependent.g. A plane in is a two-dimensional subspace.h. The nonpivot columns of a matrix are always linearly dependent.i. Row operations on a matrix A can change the linear dependence relations among the rows of A.j. Row operations on a matrix can change the null space.k. The rank of a matrix equals the number of nonzero rows.l. If an m × n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of .m. If B is obtained from a matrix A by several elementary row operations, then rank B = rank A.n. The nonzero rows of a matrix A form a basis for Row A.o. If matrices A and B have the same reduced echelon form, then Row A = Row B.p. If H is a subspace of , then there is a 3 × 3 matrix A such that H = Col A.q. If A is m × n and rank A = m, then the linear transformation is one-to-one.r. If A is m × n and the linear transformation is onto, then rank A = m.s. A change-of-coordinates matrix is always invertible.t. are bases for a vector space V , then the j th column of the change-of-coordinates matrix .

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