A First Course in Linear Algebra, 2017a

226 R n
Exercise 4.10.52 Consider the vectors of the form
⎧⎡
⎤ ⎫
⎨ 3u + v
⎬
⎣ 2w − 4u ⎦ : u,v,w ∈ R
⎩
⎭ .
2w − 2v − 8u
Is this set of vectors a subspace of R 3 ? If so, explain why, give a basis for the subspace and find its
dimension.
Exercise 4.10.53 Consider the set of vectors S given by
⎧⎡
⎤
u + v + w
⎪⎨
⎫⎪ ⎢ 2u + 2v + 4w
⎬
⎥
⎣ u + v + w ⎦
⎪⎩
: u,v,w ∈ R .
⎪ ⎭
0
Is S a subspace of R 4 ? If so, explain why, give a basis for the subspace and find its dimension.
Exercise 4.10.54 Consider the set of vectors S given by
⎧⎡
⎤ ⎫
⎨ v
⎬
⎣ −3u − 3w ⎦ : u,v,w ∈ R
⎩
⎭ .
8u − 4v + 4w
Is S a subspace of R 3 ? If so, explain why, give a basis for the subspace and find its dimension.
Exercise 4.10.55 If you have 5 vectors in R 5 and the vectors are linearly independent, can it always be
concluded they span R 5 ? Explain.
Exercise 4.10.56 If you have 6 vectors in R 5 , is it possible they are linearly independent? Explain.
Exercise 4.10.57 Suppose A is an m × n matrix and {⃗w 1 ,···,⃗w k } is a linearly independent set of vectors
in A(R n ) ⊆ R m . Now suppose A⃗z i = ⃗w i . Show {⃗z 1 ,···,⃗z k } is also independent.
Exercise 4.10.58 Suppose V,W are subspaces of R n . Let V ∩W be all vectors which are in both V and
W. Show that V ∩W is a subspace also.
Exercise 4.10.59 Suppose V and W both have dimension equal to 7 and they are subspaces of R 10 . What
are the possibilities for the dimension of V ∩W? Hint: Remember that a linear independent set can be
extended to form a basis.
Exercise 4.10.60 Suppose V has dimension p and W has dimension q and they are each contained in
a subspace, U which has dimension equal to n where n > max(p,q). What are the possibilities for the
dimension of V ∩W? Hint: Remember that a linearly independent set can be extended to form a basis.
Exercise 4.10.61 Suppose A is an m × n matrix and B is an n × p matrix. Show that
dim(ker(AB)) ≤ dim(ker(A)) + dim(ker(B)).