2. In Question 2 of the computer component of this assignment, when you enter th

Question

2. In Question 2 of the computer component of this assignment, when you enter the parameter value k 1, you generate the matrices 1 21 and C= from the matrixB1.in and matrixC1.in files. The code you are asked to produce is supposed to give the matrix A from part (b) below. So you can use your work in this question to help test your code for Question 2 (as well as the other unhidden test matrices in matrixB2.in, matrixC2. in, matrixB3.in, and matrixC3.in) (a) Explain whA |HEP- is a basis for R2, and write the standard basis vectors R as linear combinations of the vectors in (b) Suppose T: R2 ? R2 is a linear transformation satisfying Find a matrix A such that T(x) = Ax for all x E R2, and show that and are both eigenvectors of A

Explanation / Answer

(a). Let M =

1

1

1

0

-1

2

0

1

The RREF of M is

1

0

2/3

-1/3

0

1

1/3

1/3

It implies that the vectors (1,-1)T and (1,2)T are linearly independent and that the standard basis vectors (1,0)T=(2/3)(1,-1)T+(1/3)(1,2)T and (0,1)T= -(1/3)(1,-1)T+(1/3)(1,2)T, are linear combinations of these 2 vectors. Therefore, {(1,-1)T,(1,2)T } is a basis for R2.

(b). Since T is a linear transformation, it preserves both vector addition and scalar multiplication. Hence, T(1,0)T=T((2/3)(1,-1)T+(1/3)(1,2)T)=(2/3)T(1,-1)T+(1/3)T(1,2)T =(2/3)(-1,1)T+(1/3)(2,4)T = (0,2)T and T(0,1)T = T(-(1/3)(1,-1)T+(1/3)(1,2)T)=-(1/3)T(1,-1)T+(1/3)T(1,2)T = -(1/3)(-1,1)T+(1/3)(2,4)T=(1,1)T. Hence the standard matrix of T is [T(1,0)T,T(0,1)T] = A =

0

1

2

1

Further, A(1,-1)T =(-1,1)T =-1(1,-1)T and A.(1,2)T = (2,4)T = 2(1,2)T.

This implies that (1,-1)T is an eigenvector of A corresponding to the eigenvalue -1 and(1,2)T is an eigenvector of A corresponding to the eigenvalue 2.

1

1

1

0

-1

2

0

1

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