Q: 4 - 5 marks. Question 4. (5 PTS) Let a1, ..., an be distinct real numbers. Let V be the subspace of Fun(R+, R. ), where R. is the set of positive real numbers, defined through V = Spanfal, ...,x'}. Consider the transformation T := a) (2 PTS) Prove that TEC(V, V), that is T is a linear transforma- tion from V to itself; b) (2 PTS) Prove that a1, . .., an are eigenvalues of T. c) (1 PT) Prove that {xol, ..., xan} is linearly independent. (You may find the theorem proved in the Mar. 12 lecture note helpful)MATH 225 WINTER 3024 HOMEWORK 4 Question 4. (5 PTS) Let T: Mat. ..(R) - Mat. (R) be T(A) = AT. We know from Example 5 of Feb. " lecture that T is a linear transformation from Mat..(R) to itself. Calculate the eigenvalues and their associated eigenspaces. . Hint: Follow the definitions. Finding the matrix representation is not necessary. The answers may depend on the value of s. Solution. Eigenvalues, By definition AE It is an cigruvalue if and only if AT =T(A) =AA (31) for some nonzero matrix A. Now for this particular matrix A we have T(AT) = (AT) = A while at the same time T()-T(AA)-AT(A)-XA (32) Therefore (X - 1) 4=0. As A is a nonzero matrix there must hold A' = 1. Consequently A= 1 are the only candidates for eigenvalues. o A= 1 in an eigenvalue since T(/) = / where I is the identity matrix. o Am -1 is an eigemalue when n > 1 dance the matrix A with ais = -da = 1 and every other = 0 satisfies T(A) =-4. Therefore when n = 1. the only eigenvalue is 1, when n > 1 the eigenvalues are $1. Eigenspaces. For a = 1 the eigerspace associated to A= 1 is Span(( 1 )) - Mati,:(R). In the following we consider n > 1, or equivalently n 22 o A=l. We have Ti - (46 Mat.-(R)|AT - A) the subspace of symmetric matrices. o Am-1. We have T_1=(4E Mat.,.(R)| AT =-4) (33) the subspace of anti-symmetric matrices