Linear Algebra, 2020a

Section III. Basis and Dimension 143
3.30 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 ever equal?
3.31 Show that the set {(1, −1, 2, −3), (1, 1, 2, 0), (3, −1, 6, −6)} does not have the
same span as {(1, 0, 1, 0), (0, 2, 0, 3)}. What, by the way, is the vector space?
̌ 3.32 Show that
⎛
this
⎞
set of column vectors
d 1
3x + 2y + 4z = d 1
{ ⎝d 2
⎠ | there are x, y, and z such that: x − z = d 2 }
d 3 2x + 2y + 5z = d 3
is a subspace of R 3 . Find a basis.
3.33 Show that the transpose operation is linear:
(rA + sB) T = rA T + sB T
for r, s ∈ R and A, B ∈ M m×n .
̌ 3.34 In this subsection we have shown that Gaussian reduction finds a basis for the
row space.
(a) Show that this basis is not unique — different reductions may yield different
bases.
(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.
3.35 Why is there not a problem with Remark 3.15 in the case that r is bigger than
n?
3.36 Show that the row rank of an m×n matrix is at most m. Is there a better
bound?
3.37 Show that the rank of a matrix equals the rank of its transpose.
3.38 True or false: the column space of a matrix equals the row space of its transpose.
̌ 3.39 We have seen that a row operation may change the column space. Must it?
3.40 Prove that a linear system has a solution if and only if that system’s matrix of
coefficients has the same rank as its augmented matrix.
3.41 An m×n matrix has full row rank if its row rank is m, and it has full column
rank if its column rank is n.
(a) Show that a matrix can have both full row rank and full column rank only if
it is square.
(b) Prove that the linear system with matrix of coefficients A has a solution for
any 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 its
matrix of coefficients A has full column rank.
(d) Prove that the statement “if a system with matrix of coefficients A has any
solution then it has a unique solution” holds if and only if A has full column
rank.
3.42 How would the conclusion of Lemma 3.3 change if Gauss’s Method were changed
to allow multiplying a row by zero?
3.43 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), and
rank(A + B)?