Matrix: please show all your works

(1) Dene T : R4 A. R7 by 23 a: + y y + 3 z + w 32; 4w 3:1: 2: _ '9' Verify that T(a.u + be) = aT(a} + bT(v) where a, v E R4 and a, b E R are arbitrary. (Note: It is not acceptable to first nd the matrix of T and then verify linearity.) Now nd the matrix of this linear map with respect to the standard bases for R4 and R7. (2) Suppose that S is a linear map from R2 to R4 and that Swansea 1 1 1 S(s)= -1 1 while 1 3 2 S( . )= -3 1 Find the matrix of S with respect to the standard bases for R2 and R4. (3) Let F4 be the set of all polynomials of degree 4 with real coefcients. Note that we can identify 9'4 with IRE by: {343:4 + {13.7.3 + a2m2 + mm + a0 I> :12 Similarly identify the set 93 of polynomials of degree 3 with R4. Dene D : #4 > 9'3 by Dp(:r) = 10"(33). We know from calculus I that D is linear. Find the matrix of D