Problem Set 1
due at 1pm on March 09, 2023
Directions: Justify all answers, show all solutions, and work independently. To get full credit, proofs
and explanations should be clear, coherent, complete, and concise; and examples should be verified
to satisfy prescribed properties.
1. Observe that Cn is a vector space over R under
(z1 , . . . , zn ) + (w1 , . . . , wn ) = (z1 + w1 , . . . , zn + wn )
r(z1 , . . . , zn ) = (rz1 , . . . , rzn )
for all r ∈ R and for all zj , wj ∈ C. Let
U := {(z1 , z2 , z3 ) ∈ C3 | z2 = −z1 } .
(a) Show that U is a subspace of C3 over R.
(b) Find a basis for U over R.
(c) Give examples of subspaces V and W of C3 over R such that
V 6= W and C3 = U ⊕ V = U ⊕ W .
2. (a) Write down your student number and use its nine digits to come up with an ordered basis
B 0 for R3 over R.
(b) If B = {e1 , e2 , e3 }, what is m(idR3 )B,B 0 ?
3. Let V , W , and U be vector spaces over F such that V and W are finite dimensional.
Let f : V → W and g : W → U be linear transformations over F.
Let h : f (V ) → U such that h(w) := g(w) for all w ∈ f (V ).
(a) Show that h is a linear transformation over F
How are Im(h) and Im(g ◦ f ) related, and why?
How are ker(h) and ker(g) related, and why?
(b) Show that rank(f ) ≤ rank(g ◦ f ) + ν(g).