Problem 4: (7 points)
Let T: R2[x]?R2[x] be a given linear operator defined by
T(a0 + a1x + a2x²) = a2x².
Find the kernel of T.
A) < (0,1,0) >
B) <1,x>
C) <1,x²>
D) <x,x²>
E) <1,x,x²>
Problem 5: (7 points)
Find the determinant of the matrix $\begin{pmatrix} a & a & a\\a & b & b\\a & b & c \end{pmatrix}$.
A) 0
B) a
C) a(b-a)(b-c)
D) a(c-a)(c-b)
E) a(b-a)(c-b)
Problem 6: (7 points)
Given A, B, C ? Mnxn(R), which one of the following statements is false:
A) If ? is an eigenvalue of A with associated eigenvector v then v is an eigenvector of A², associated with the eigenvalue ?².
B) The determinant of ABC is |A||B||C|.
C) M1M2 is orthogonal if M1 and M2 are both orthogonal matrices.
D) The determinant of In + A is 1 + det A.
E) det A ? 0 if and only if A is invertible.
