Let ?={w?1,w?2,?,w?n} be an ordered basis for ?n. We already know that the coord

Question

Let ?={w?1,w?2,?,w?n} be an ordered basis for ?n. We already know that the coordinates for v???n relative to ? is the ordered n-tuple (c1,c2,?,cn) with v?=c1w?1+c1w?1+?+cnw?n. We can go a step further and create matrices relative to a basis.

If T:?n??n is a linear transformation then we create a matrix [T]? by having the (i,j)-entry of the matrix be the jth coordinate of T(w?i) relative to ?.

(c) Determine the characteristic polynomials for both your matrices in (b).

(d) Show that your matrices in (b) are similar.

Explanation / Answer

1

5

0

1

2

0

1

9

1

0

-1

9

The RREF of A is

1

0

0

6

0

1

0

-1

0

0

1

-3

Now it is apparent that (1,9,9)T = 6(1,2,1)T-1(5,0,0)T-3(0,1,-1)T so that the coordinates of (1,9,9)T relative to the given basis are (6,-1,-3)T.

(b), (c ) and (d). The meaning of x???2 , d?=[31] and [T]? are not clear. Please clarify.

1

5

0

1

2

0

1

9

1

0

-1

9

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