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I.D. number................................................................................

Math 2270-1

Sample Final Exam

December 8, 2000

This exam is closed-book and closed-note. You may not use a calculator which is capable of doing linear algebra computations. In order to receive full or partial credit on any problem, you must show all of your work and justify your conclusions. There are 200 points possible, and the point values for each problem are indicated in the right-hand margin. Of course, this exam counts for 30% of your final grade even though it is scaled to 200 points. Good Luck!

1) Let

Then L( x )=A x is a matrix map from R ^3 to R ^2.

1a) Find the four fundamental subspaces associated to the matrix A.

(20 points)

1b) State and verify the theorem which relates rank and nullity of A, in this particular case

(5 points)

1c) Find an orthonormal basis for R^3 in which the first two vectors are a basis for the rowspace of A and the last vector spans its nullspace.

(10 points)

1d) Find an implicit equation ax+by+cz = 0 satisfied for precisely those points which are in the rowspace of A.

(5 points)

2a) Exhibit the rotation matrix which rotates vectors in R^2 by an angle of a radians in the counter-clockwise direction.

(5 points)

2b) Verify that the product of an a -rotation matrix with a b -rotation matrix is an ( a + b )-rotation matrix.

(10 points)

3) Find the least-squares line fit through the four points (1,1), (2,1), (-1,0), (-2,0) in the plane.

(10 points)

4a) Explain the procedure which allows one to convert a general quadratic equation in n-variables

into one without any ``cross terms''. Be precise in explaining the change of variables, and the justification for why such a change of variables exists.

(10 points)

4b) Apply the procedure from part (4a) to put the conic section

into standard form. Along the way, identify the conic section.

(20 points)

5) Let

5a) Find the inverse of C using elementary row operations.

(15 points)

5b) Find the inverse of C using the adjoint formula.

(15 points)

6a) Define what it means for a function L:V-->W between vector spaces to be a linear transformation .

(5 points)

6b) Define what it means for a set S={ v 1 , v 2 , . . . v n } to be a basis for a vector space V.

(5 points)

6c) For a linear map L as in part (6a), define the nullspace (kernel) of L.

(5 points)

6d) Let S={ v 1 , v 2 , . . . v n } be a basis for V, and T={ w 1 , w 2 , . . . w m } be a basis for W. Let L:V-->W be linear. Explain what the matrix for L with respect to S and T is, and how to compute it.

(10 points)

7) Let

7a) Let

(Note these are the columns of your matrix C from #5.) Let L(x)=Ax, a matrix map from R^3 to R^3. Thus A is the matrix of L with respect to the standard basis of R^3. Find the matrix of L with respect to the T basis.

(20 points)

7b) Let

Compute L(v) two ways: once using the matrix for L with respect to T, and once using the matrix A. Verify that your answers agree.

(10 points)

8) True-False. Two points each. No justification required!

(20 points)

8a) Let A be an n by n matrix. Then if Ax=Ay it follows that x=y.

8b) If A and B are n by n matrices, then

8c) If the vectors v 1 , v 2 , . . ., v k are orthogonal to w , then any vector in the span of { v 1 , v 2 , . . ., v k } is also orthogonal to w .

8d) Every set of five orthonormal vectors in R ^5 is automatically a basis for R ^5.

8e) The number of linearly independent eigenvectors of a matrix is always greater than or equal to the number of distinct eigenvalues.

8f) If the rows of a 4 by 6 matrix are linearly dependent then the nullspace is at least three dimensional.

8g) A diagonalizable n by n matrix must always have n distinct eigenvalues.

8h) Every orthogonal matrix is diagonalizable.

8i) If A and B are orthogonal matrices then so is AB.

8j) If A is a square matrix and A^2 is singular, then so is A.