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(b) State a formula relating the matrices in part (a).
(c) Verify that the matrices in part (a) satisfy the formula you stated in part (b).
Show that if is the zero transformation, then the matrix of T with respect to any bases for V and W is a zero
14. matrix.
Show that if is a contraction or a dilation of V (Example 4 of Section 8.1), then the matrix of T with respect to
15. any basis for V is a positive scalar multiple of the identity matrix.
Let be a basis for a vector space V. Find the matrix with respect to B of the linear operator
16. defined by ,,,.
Prove that if B and are the standard bases for and , respectively, then the matrix for a linear transformation
17. with respect to the bases B and is the standard matrix for T.
18. (For Readers Who Have Studied Calculus)
Let be the differentiation operator . In parts (a) and (b), find the matrix of D with respect to the
basis .
(a) , ,
(b) , ,
(c) Use the matrix in part (a) to compute .
(d) Repeat the directions for part (c) for the matrix in part (b).
19. (For Readers Who Have Studied Calculus)
In each part, is a basis for a subspace V of the vector space of real-valued functions defined on the real
line. Find the matrix with respect to B of the differentiation operator .
(a) , ,
(b) , ,
(c) , ,
