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Answer: The required answers are
(a) T is proved to be a linear transformation.
(b) The matrix A such that T(x) = Ax is
Step-by-step explanation: We are given a linear transformation T : R³ → R² defined as follows :
We are to
(a) prove that T is a linear transformation
and
(b) find a matrix A such that T(x) = Ax.
(a) Let s, t are any real numbers and (a, b, c), (a', b', c') ∈ R³.
Then, we have
So, we get
Therefore, T is a linear transformation.
(b) We know that B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is a standard basis for R³ and B' = {(1, 0), (0, 1)} is a standard basis for R².
So, we have
So, the matrix A such that T(x) = Ax will be given by
Thus,
(a) T is proved to be a linear transformation.
(b) The matrix A such that T(x) = Ax is