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Six practice questions on linear transformations and matrices, covering topics such as finding standard matrices, determining one-to-one and onto transformations, finding vectors in the image of a transformation, and finding the inverse of a matrix. Students preparing for exams on linear algebra will find these questions useful for review and practice.
Typology: Exams
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Define the linear transformation T : R^3 Ï R^3 so that
x 1 x 2 x 3
x 1 + 2x 3 2 x 1 + x 2 + 3x 3 x 1 − x 2 + 3x 3
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^3 Ï R^3 and S : R^3 Ï R^3 so that
T
x 1 x 2 x 3
x 1 − 2 x 2 + x 3 x 2 − x 3 2 x 3
(^) and S
x 1 x 2 x 3
− x 3 − x 2 + 2x 3 − x 1 + 2x 2 − x 3
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 determine the value(s) of λ for which the matrix
λ − 1 0 − 1 λ − 1 0 − 1 λ
is invertible.
Let T : R^3 Ï R^3 be a linear transformation and B = { v⃗ 1 v,⃗ 2 v,⃗ 3 } a basis for R^3. Suppose Tv(⃗ 1 ) = ( − 2 , 1 , 1), Tv(⃗ 2 ) = (0, 1 , − 1) and Tv( ⃗ 3 ) = ( − 2 , 2 , 0). a.) Determine whetherw⃗ = ( − 6 , 5 , 0) is in the range of T. b.) Find a basis for the range of T. c.) Find a basis for the null space of T.(Remark. Range of T is the same space as the column space of A where A is the standard matrix of T.)
