(1) (3 points) Set u = 0 8 and set v = AOO (a) Compute the linear combination -2u + 5v. (b) Compute the dot product u . v. In problems (2) (3) below we use the standard definition of a "linear trans- formation", which is as follows: . Definition: A function T : Rm - R" is a linear transformation if the following two properties hold for any vectors u, v E RM and any number c. (i T(utv) = T(u) +T(v) (ii) T(cu) = cT(u) Note that Theorem 2.1.3 in our text book states that the above definition of "linear transformation" is equivalent to Definition 2.1.1 in our text book. (2) (3 points) Let T, u, v be as in the definition above, and let a, b be real numbers. Using only properties (i) (ii) in the definition given above, show that T(au + bv) = aT(u) + bI(v)(3) (4 points) Let T : R2 - R3 denote linear transformation which satisfies O (a) Find numbers a, b, c, d such that a 3 ]to [ ] = =] and ] + @ [ 1 ] =ez (b) Find the matrix A for T (i.e. the matrix A such that Ax = T(x) for all x ( R2). Hint: use (2),3(a) above and Theorem 2.1.2 in the text book