MATH 304 Midterm: MATH 304 Binghamton Math304 Fall2018 Exam1V1 Solutions

Let P denote the vector space of all polynomials and f'(x)denote the derivative of all numbers in P. State true or false tothe following:

1. T: R^2 --> R^3 defined by T(x,y)=(?^2x,0, y/2) is a linear transformation

2. T: R^2 --> R^3 defined by T(x,y)=(x+y, x-y, xy) is alinear transformation

3. T: R^2 --> R^3 defined by T(x,y)=(1,0,0) is a lineartransformation

4. T: R^2 --> R^3 defined by T(x,y)= (y, x, y) is a lineartransformation

5. T: R^2 --> R^3 defined by T(x,y)= (x, x^2, x^3) is alinear transformation

6. T: R^2 --> R^3 defined by T(x,y)= (2x+3y, 3x+4y, 4x+5y) isa linear transformation

7. T: P --> P defined by T(f(x))= f(0) +f'(0)x + f''(0)x^2 isa linear transformation

8. T: P --> P defined by T(f(x))= f(1) + f'(2) + f''(3)x^2 isa linear transformation

9. T: P --> P defined by T(f(x))= f(x-3) is alinear transformation

10. T: P --> P defined by T(f(x))= f(x)-3 is a lineartransformation

K(x,y,z) = (x, x + y,x + y + z)

L(x,y,z) = (2x - y, x + 2y)

S(x,y,z) = (z,y,x)

T(x,y) = (2x + y,x + y,x - y,x - 2y)

G: R^3 -> M(2,2) defined by G(x,y,z) = (y z , (-x + y) -x)

H: M(2,2) -> P2 defined by H(a b, c d) = (a + b) + (a - 2d)x+ dx^2

Find the indicated linear transformation if it is defined. ifnot defined explain why it is not:

1) K + S

2) GS

3)2S^3 - S^2 + 3S - 4I

Find the matrix that represent each of the linear transformationK,L,S,and T for the following

1)TL

2)K + S

Find the matrix that represent each of linear transformation intwo ways, first using the result normal way and using therepresention of the matrix.

1)

T:R^3 --> R^4 given by T(X) = Ax, where

A = [1 1 0

2 0 3

-1 1 1

1 1 2]

and v1 = (-3,3,2) v2 = (0,0,0)

2) T: R^4 --> R given by T(x1,x2,x3,x4) = x1 + 2x2 + 3x3 +4x4 and v1 = (1,2,1,-2) v2 = (2,3,0,-2)

3)T: M(2,3) --> P2

P2 = (a11 + a13)x^2 + (a21 - a22)x + a23

v1 = [-1 3 1

2 2 0]

v2 = [2 1 -2

1 1 0]

& determine wheter w1 or w2 is the iamge of thecorresponding linear transformation

w1 = x^2 + 2x + 1,w2 = x - 2.