Math 340 Spring 08Lecture 4 G.MeyerExam 21. A matrix A has row vectors α1, α2, α3, α4and column vectors β1, β2, ..., β6.If A has reduced row echelon form1 0 3 −2 0 30 1 −4 5 0 20 0 0 0 1 10 0 0 0 0 0Answer the following questions (your answer may use the αs and βs.a) A basis for the null space of A is:b) A basis for the row space of A is:c) A basis for the column space of A is:d) What is the rank of A?2. Consider the set S = {111,110,101} and T = {112,121,011}a) Verify in any legitimate way you can that S and T are bothbases of R3.b) Determine the transition matrix PS←Tfrom the T basis to theS basis.c) If [v]T=123, determine [v]S.3. Answer the following questions as True (T) or False (F) by circling Tor F below. No justification wanted.a (T) (F) If u1, u2, u3, u4, u5are linearly independent vectors in a5-dimensional subspace U of R8, then they are a basis of U.b (T) (F) If v1, v2, v3, v4, v5span a 5-dimensional subspace V of R8,the n they are a basis of V .1c (T) (F) If A is a 6 by 8 matrix, then you can be sure that thehomogeneous system Ax = 0 has a nontrivial solution.d (T) (F) A linearly independent set of 3 vectors in a 5-dimensionalvector space V can always be enlarged to a basis of V .e (T) (F) A set of 8 vectors in a 6-dimensional vector space Ualways contains 6 vectors that form a basis of U.f (T) (F) The set of all singular 4 by 4 matrices is a subspace ofthe vector space M4,4of all 4 by 4 matrices.g (T) (F) A 3 by 5 matrix could have rank 4.h (T) (F) If the nullity (dimension of nullspace) of the 5 by 5 matrixA is 0, then A is singular.i (T) (F) If A and B are 4 by 6 matrices with the same row space,then A and B have the same column space.j (T) (F) If A is a 4 by 5 matrix and B is a 4 by 6 matrix and theircolumn spaces have the same dimension, then their row spaces havethe same dimension.k (T) (F) If v1, v2, v3, v4span a subspace V of a vector space U ,and v1is a linear combination of v2, v3, v4, then v2, v3, v4span V .l (T) (F) The set of all real polynomials of degree ≤ 4, using stan-dard addition and scalar multiplication, is a vector space of dimension4.m (T) (F) The set of all 3 by 3 symmetric matrices forms a subspaceof dimension 6 of the vector space M3,3of 3 by 3 matrices.4. Prove: If A is a nonsingular n by n matrix, and v1, v2, v3, v4are lin-early independent vectors in Rn, then Av1, Av2, Av3, Av4are linearlyindependent vectors in