Linear Algebra
-1
Consider the map ⊳⋅≀∙⇠∙⋅⊽∙∙ ∶ C3[t] −≻ (3“ given by p
I’ll)
[Remember that C3[t] is the space of polynomials of degree no larger than 3 with complex
oefficients.)
(a) Show that ⋅≀∙⇠∙⋅⊽∼∙ is C-linear. Begin your answer by stating the condition(s) you have to
check.
(b) (i) Show that the set
13: {2113+ 1, t3+t- 1, t2- 1, t3 -t} ⊆ C3[t]
is linearly independent. Begin your answer by stating the condition(s) you have
to check.
(ii) Show that 1’3 is a basis of C3[t].
(c) Compute the basis representation [11:]; of ⊳↴≀∙⇠∙⋅⊁∙ with respect to the basis 1’3 of mm and
1 0 1
(1 (l (1}
0 0 1
of C“.
You do not need to show that ∧∫⋅ is a basis of C“.
(d) ls ⋅≀∙⇠∙⋅⊽∼∙ injective? Justify your answer.