Linear Algebra

130 Chapter Two. Vector Spaces3.27 Give an example to show that, despite that they have the same dimension,the row space and column space of a matrix need not be equal. Are they everequal?3.28 Show that the set {(1, −1, 2, −3), (1, 1, 2, 0), (3, −1, 6, −6)} does not have thesame span as {(1, 0, 1, 0), (0, 2, 0, 3)}. What, by the way, is the vector space?̌ 3.29 Show that this set of column vectors{( ) }d1∣∣ 3x + 2y + 4z = d 1d 2 there are x, y, and z such that x − z = d 2d 3 2x + 2y + 5z = d 3is a subspace of R 3 . Find a basis.3.30 Show that the transpose operation is linear:(rA + sB) trans = rA trans + sB transfor r, s ∈ R and A, B ∈ M m×n,̌ 3.31 In this subsection we have shown that Gaussian reduction finds a basis forthe row space.(a) Show that this basis is not unique — different reductions may yield differentbases.(b) Produce matrices with equal row spaces but unequal numbers of rows.(c) Prove that two matrices have equal row spaces if and only if after Gauss-Jordan reduction they have the same nonzero rows.3.32 Why is there not a problem with Remark 3.14 in the case that r is biggerthan n?3.33 Show that the row rank of an m×n matrix is at most m. Is there a betterbound?̌ 3.34 Show that the rank of a matrix equals the rank of its transpose.3.35 True or false: the column space of a matrix equals the row space of its transpose.̌ 3.36 We have seen that a row operation may change the column space. Must it?3.37 Prove that a linear system has a solution if and only if that system’s matrixof coefficients has the same rank as its augmented matrix.3.38 An m×n matrix has full row rank if its row rank is m, and it has full columnrank if its column rank is n.(a) Show that a matrix can have both full row rank and full column rank onlyif it is square.(b) Prove that the linear system with matrix of coefficients A has a solution forany d 1 , . . . , d n ’s on the right side if and only if A has full row rank.(c) Prove that a homogeneous system has a unique solution if and only if itsmatrix of coefficients A has full column rank.(d) Prove that the statement “if a system with matrix of coefficients A has anysolution then it has a unique solution” holds if and only if A has full columnrank.3.39 How would the conclusion of Lemma 3.3 change if Gauss’ method is changedto allow multiplying a row by zero?̌ 3.40 What is the relationship between rank(A) and rank(−A)? Between rank(A)and rank(kA)? What, if any, is the relationship between rank(A), rank(B), andrank(A + B)?