1. Let 3-12 (a) (8 pts) Give an explanation that refers to the definition of a b
ID: 3169425 • Letter: 1
Question
1. Let 3-12 (a) (8 pts) Give an explanation that refers to the definition of a basis as to why B is a basis for R (b) (3 pts) Find x relative to the standard unit normal basis, where3.using the definition of the coordinates of x relative to the basis B (e) (6 pts) Find the change-of-coordinates matrix from B to E (the standard unit normal basis). where: (d)(3pts)Useyour resultfrom(c)tofind (e) (6 pts) Find the change of coordinates matrix from to B. (You may use your (0) (3 pts) Use your result from (e) to find for4 IBExplanation / Answer
The RREF of A is I3 which implies that the vectors in B are linearly independent and span R3. Hence B is a basis for R3.
(b). We have [x]B = (-2,3,1)T = -2(2,-1,3)T+3(1,0,-1)T+1(3,2,1)T = (-4,2,-6)T+(3,0,-3)T+(3,2,1)T= (2,4,-8)T. Thus, the coordinates of x with respect to the standard unit normal basis are (2,4,-8)T.
(c ). The change of coordinates matrix from B to is A =
(d). If [x]B = (-3,2,2)T, then x = A[x]B = (2,7,-9)T
(e ). Let M =
The RREF of M is
This implies that the change of coordinates matrix from to B is P=
(f). If x = (2,4,-8)T, then [x]B = Px = (-2,3,1)T.
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