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**preview**shows page 1. to view the full**4 pages of the document.**MATH 240 – Fall 2012 – Exam 2

CALCULATORS ARE NOT ALLOWED

TURN OFF ALL ELECTRONIC DEVICES

.

*** THERE ARE QUESTIONS ON BOTH SIDES OF THIS PAPER ***

Answer each question on a separate sheet of paper. Use the back side if necessary. On each sheet, put your

name, your section leader’s name and your section meeting time. No proof is needed for TRUE-FALSE

questions; just write clearly. You may assume given matrix expressions are well deﬁned (i.e. the matrix sizes

are compatible).

1. (a) (8 pts.) The vertices of a triangle Tin R2are the points (13,1),(15,3),(20,6).

Compute the area of T.

Solution.

Subtract (13,1) from each vector to put one corner on the orgin. The other two

corners become (2,2),(7,5). The area is (1/2) times the absolute value of the de-

terminant of the matrix with these two vectors as columns, so area(T) = 2.

(b) (8 pts.) Let Sbe the triangle in R2with vertices (100,100),(100,101),(101,100).

Let B= ( 7 5

5 4 ). Compute the area of BS ={Bx :xis in S}.

Solution.

area(BS) = |det(B)|area(S) = (3)(1/2) = 3/2 .

(c) (9 pts.) Suppose Ais a m×nmatrix and R= rref(A).

Answer the following TRUE or FALSE.

i. The column space of Aequals the column space of R.

FALSE.

ii. The row space of Aequals the row space of R.

TRUE.

iii. The null space of Aequals the null space of R.

TRUE.

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