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10 Nov 2019

Is this transformation linear?


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


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Irving Heathcote
Irving HeathcoteLv2
14 May 2019

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Please show your work step by step and answer allquestions to get full points:


Which of the following transformations are linear?


1) T : R^3 to R^2 defined by T(x, y) = (xy, 0, x, y)



2) T : R^3 to R defined by T(x, y, z) = x - 3y+ 2z



3) T : R to R definedby T(x) = (-3)^x



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

consider the transformation T: R^3--->R^3 defined byT(x,y,z)=(-z,y+z,2z-x)


a) Find the standard matrix for T and use it to find the imageof (2,3,-1)

b) Is T an invertible transformation?

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