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Material Type: Exam; Professor: Schurle; Class: MATRICES/MATRIX CALCULATNS-C S; Subject: Mathematics; University: University of Texas - Austin; Term: Fall 2010;
Typology: Exams
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? If so, find a basis for its
eigenspace. If not, justify your answer.
T (b 1 ) =
and T (b 2 ) =
(a) (6 points) Calculate and simplify T (4b 1 − 2 b 2 ).
(b) (6 points) What is the matrix for T relative to the basis B and the standard basis for R^3?
(c) (6 points) Suppose C is another basis for V , where
c 1 = 5b 1 + 2b 2 ,
c 2 = 7b 1 + 3b 2. What is the matrix for T relative to the basis C and the standard basis for R^3?
u 1 =
, u 2 =
, u 3 =
.
(a) (4 points) Verify that {u 1 , u 2 , u 3 } is an orthogonal basis for W.
(b) (10 pts.) Write y =
as the sum of a vector in W and a vector in W ⊥.
(c) (4 points) Find the distance from y to the subspace W.
(d) (6 points) Find a basis for W ⊥.
