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Six sample exam questions on linear transformations and matrices. The questions cover topics such as finding standard matrices, determining if transformations are one-to-one or onto, finding vectors that map to given vectors, and calculating the inverse of a matrix. Students are expected to use matrix operations and the invertible matrix theorem to solve the problems.
Typology: Exams
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Define the linear transformation T : R^2 Ï R^3 so that [ (^) x 1 x 2
x 1 − x 2 − x 1 + 2x 2 x 1 + x 2
a.) Find the standard matrix of T. b.) Is T one-to-one? c.) Is T onto?
d.) If there is any, find a vectorv ⃗ such that Tv(⃗ ) = ⃗b where b⃗ =
Define the linear transformations T : R^2 Ï R^3 and S : R^3 Ï R^2 so that
T
([ (^) x 1 x 2
x 1 − 2 x 2 3 x 1 + x 2 2 x 2
(^) and S
x 1 x 2 x 3
[ (^) x 1 +^ x 2 x 1 − x 2
a.) Find the standard matrix of S ◦ T. b.) Find the standard matrix of T ◦ S. c.) Find, if there is any, a vectorv⃗ such that (S ◦ T)(v⃗ ) = ⃗b where b⃗ =
Use the invertible matrix theorem to find the value(s) of s so that the matrix
s 1 0 0 s 1 0 1 s
is invertible.
Define the linear transformation T : R^4 Ï R^3 by T
x 1 x 2 x 3 x 4
x 1 + x 2 x 2 − x 3 x 1 + x 4
a.) Find the column space of T. b.) Find the null space of T. c.) Find a basis for the column space of T. d.) Find a basis for the null space of T.(Remark. The column space of T is Col A and the null space of T is Nul A where A is the standard matrix of T.)
