Find T(v) by using the matrix relative to B and B'. T: R2 → R3, T(x, y) = (x + y, x, y), v = (7, 4), B = {(1, −1), (0, 1)},B' ={(1, 1, 0),(0, 1, 1),(1, 0, 1)}

Find T(v) by using (a) the standard matrix and (b) the matrix relative to B and B'. T: R2 -> R3 T(x, y) = (-x, y, x+y), v = (0, 1), B = {(1, 1), (1, -1)}, B' = {(0, 1, 0), (0, 0, 1), (1, 0, 0)}

Find T(v) by using (a) the standard matrix and (b) the matrix relative to B and B′.T: R2→R3, T(x, y) = (−x, y, x + y), v = (0, 1), B = {(1, 1), (1, −1)}, B′ = {(0, 1, 0), (0, 0, 1), (1, 0, 0)}

Find the value(s) of λ for the matrix D to be singular. D=(1,2,0 0,λ + 1,2 0,1,λ)

Find the standard matrix for the linear transformation T.T(x, y, z) = (3z − 2y, 4x + 11z)

Is T(x,y,t) = xcos(t) + ysin(t) linear, and how would you rewrite it using dot product and also using matrix notation?

Find T(v) by using (a) the standard matrix and (b) the matrix relative to B and B′.T: R4→R2, T(x1, x2, x3, x4) = (x1 + x2 + x3 + x4, x4 − x1), v = (4, −3, 1, 1), B = {(1, 0, 0, 1), (0, 1, 0, 1), (1, 0, 1, 0), (1, 1, 0, 0)}, B′ = {(1, 1), (2, 0)}

Find T(v) by using (a) the standard matrix and (b) the matrix relative to B and B′.T: R3→R3, T(x, y, z) = (x + y + z, 2z − x, 2y − z), v = (4, −5, 10), B = {(2, 0, 1), (0, 2, 1), (1, 2, 1)}, B′ = {(1, 1, 1), (1, 1, 0), (0, 1, 1)}

Find T(v) by using (a) the standard matrix and (b) the matrix relative to B and B′.T: R3→R2, T(x, y, z) = (x − y, y − z), v = (2, 4, 6), B = {(1, 1, 1), (1, 1, 0), (0, 1, 1)}, B′ = {(1, 1), (2, 1)}

Find the standard matrix for the linear transformation T.T(x, y) = (5x + y, 0, 4x − 5y)