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Unformatted text preview: Name: Instructor: E] Fengbo HANG El Max LIEBLICH I] Fabrice ORGOGOZO Linear algebra with applications (MAT 202)
Midterm (Fall 2005)
Time: 90 minutes z Books, notes, calculators are not permitted on this exam. Unless otherwise explicitly written, you have to explain all your answers: a correct answ
without explanation will never be granted more than half the credit. ' Special attention should be given to the clarity of your explanations and the accuracy 0 your computations. Both will be taken into account. ' As required by the University, please write out, Sign and respect the following pledge: I pledge my honor that I have not violated the honor code during this examination. 1. (a) Find a matrix B such that AB = Idg, where " 1 23
A=(—1 3 0) (b) Is A invertible '2' 3. (a) Give an example of a linear transformation T whose kernel is the line spanned by
1
1
1
1 and write its matrix in the standard basis.
(b) What is the rank of T? 2. True or False (a) If Q is an invertible matrix, and A a matrix such that AQ is deﬁned, then rank (AQ) = mill(£1. (b) If m, 112, m, U4 are linearly independent vectors in R“, then “.1 — Zug, til + €112,113 + H4, 21:3 + H4
are linearly independent too. 4. Let (a) Find a. basis for Im(A).
(b) Find a basis for Ker(A). 5. Let E be the following system of equations, where k is a real number: 3: + 23: + 22 = 5 23: + 23; + 32 = 5 3x+4y+(k2+1)z = 8+k
“'(a) For which values of k is this system consistent? (b) For the values of k such that E has inﬁnitely many solutions, give a parametric description of
those solutions. 6. (a) Let P be a plane in R3 and T the orthogonal projection on P. Explain why there exists a
basis $3 of R3 such that the Eimatrix of T is 1
0
0 (Dr—‘0
(DOC) (b) For P = {(x,y,z) E R3, such that 2005s + 20063; + 20072 = 0}, ﬁnd such a basis B. 7. (a) Show that it is not possible to ﬁnd four vectors 91, 1.12, vans in the plane R2 such that for all
1 S 2' < j S 4, 'U“  0:; < D (i.e. the angle between two different vectors is obtuse).
Hint: Argue geometrically. (b) Let us assume now that an, wg, 103, 104, 105 are ﬁve vectors in R3, such that that for all
1 g 1' < j S 5, w;  w: < 0. For each 1 s t' S 4, write 1:), = 1);: + Aims, where vgiws, i.e. v; w belongs to the plane Span(w5)i. i. Show that each A; is strictly negative.
ii. Showthatforall 1 $t<j $4, <0.
Hint: Expand the dot product + Aims.)  + Ajw5). Remark: It follows from part (a) that in fact such a collection of vectors (1001935 cannot exist
in R3. ...
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 Spring '08
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 Linear Algebra, Algebra, El Max LIEBLICH, Fabrice ORGOGOZO

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