MATH 3114 - Linear Algebra: Koofer
Due on
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Section 2.1: #5
Let T:P2(R)→P3(R) be defined by T(f(x)) = xf(x) + f0(x).
a. Prove that T is a linear transformation.
b. Find bases for both N(T)and R(T), and computer the nullity and rank of T. Verify the
dimension theorem.
c. Determine whether T is one-to-one or onto.
Section 2.1: #15
Prove that T:P(R)→P(R)defined by T(f(x)) = Rx
0f(t)dt is linear, one-to-one, and not onto.
Section 2.1: #21
Let Vbe the vector space of sequences. Define the functions T, U :V→Vby:
T(a1, a2, . . . ) = (a2, a3, . . . )and U(a1, a2, . . . ) = (0, a1, a2, . . . ).
a. Prove that Tand Uare linear
b. Prove that T is onto, but not one-to-one
c. Prove that U is one-to-one but not onto
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