SAMPLE MIDTERM II
(1) Let T1:V→Wand T2:W→Xbe two linear transformations. Assume
that T2T1is one-to-one. Show that T1is also one-to-one.
(2) Let Abe a square matrix with A2= 0. Show that Ais NOT invertible.
(3) Let T:V→Vbe an invertible and diagonalizable linear transformation.
Show that T−1is also diagonalizable.
(4) Let T:P2(R)→P2(R) be the linear transformation T(f(x)) = f(x+ 1).
Thus T(1) = 1 and T(x2) = x2+ 2x+ 1. Is Tdiagonalizable? You must
justify your answer.
(5) Let Vbe a real vector space of dimension 4. A linear transformation
T:V→Vhas characteristic polynomial t4+t2. Is Tdiagonalizable?
Justify your answer.
(6) Let T:V→Vbe a linear transformation. Let λand λ0be two different
scalars. If Eλand Eλ0are the two eigenspaces of Tcorresponding to λand
λ0, show that Eλ∩Eλ0={0}.
(7) Find the general solution of the system of differential equations:
y0
1=y1+ 3y2
y0
2=y1−y2.
(8) Let Mbe the 2 ×2 matrix µ1a
a−1¶, where ais a real number. Prove that
Mis diagonalizable.