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April 28th, 2009
Math 251
Lu´ıs Finotti Spring 2009
Name:.........................................
Student ID (last 6 digits): XXX-.................
Final
You have two hours to complete the exam. Do all work on this exam, i.e., on the page of the respective assign- ment. Indicate clearly, when you con- tinue your solution on the back of the page or another part of the exam. Write your name and the last six digits of your student ID number on the top of this page. Check that no pages of your exam are missing. This exam has 7 questions and 12 printed pages (in- cluding this one and a page for scratch work in the end). No books, notes or calculators are al- lowed on this exam! Show all work! (Unless I say oth- erwise.) Even correct answers with- out work may result in point deduc- tions. Also, points will be taken from messy solutions. Good luck!
Question Max. Points Score
Total 100
- [10 points] Put the following matrix in reduced row echelon form:
(c) [4 points] Let T : R^3 → R^4 be the linear transformation given by
T (x 1 , x 2 , x 3 ) = (2x 1 − 3 x 2 , 0 , x 2 − x 3 , x 1 ).
Give [T ] [i.e., the matrix associated to this linear transformation].
(d) [4 points] If W = span({(1, 2 , − 3 , 1), (0, 2 , 0 , 2), (− 1 , 1 , 3 , 4)}), then the orthogonal com- plement of W given by what matrix space [i.e., row space, column space, or null space] of the what matrix?
(e) [4 points] Let TA be the a linear transformation associated to the m by n matrix A. If TA is one-to-one, then what can we say about the rank of A? [If this rank is unrelated to whether or not TA is one-to-one, just say so.]
(f) [5 points] Let A be a 3 by 3 matrix with eigenvalues −2 and 1, with their respective eigenspaces being span{(1, 0 , −1), (1, 1 , 1)} and span{(0, 0 , 1)}. Give the matrix P such that P −^1 AP is diagonal, as well as P −^1 AP itself.
- [10 points] Let v be a vector in Rn. Show that the set W of all vectors w in Rn^ such that v · w = 0 is a subspace of Rn. [Note: Part of this is to show that W is non-empty. To show this you just need to find a vector that you can guarantee is in W .]
- [10 points] Let S = { 1 , x, x^2 } and S′^ = {1 + x, 1 + x^2 , x + x^2 }. [Both are bases of P 2 .] Give the transition matrix from S to S′.
- Let
A =
Then we have:
A −−row red.−−−−−→
and^ A
T (^) −−row red.−−−−−→
You do not need to justify any of the items below.
(a) [5 points] Give the rank of A and the dimensions of the row space and of the column space of A?
(b) [5 points] Find a basis for the row space of A made of rows of A.
(c) [5 points] For each row of A not in the basis of the previous item, give its coordinates with respect to the basis you found.
(d) [5 points] Which vectors from the standard basis of R^6 you can add to the vectors in the basis of the row space you found above to obtain a basis of all of R^6?
