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Algebra
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If p > 5 is prime, and p is divided by 10 (that is, p = 10q + r, with 0 ≤ r < 10), show that the only possible remainders are r = 1,3,7,9.

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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


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