Math 2270-1

Review, and Practice Exam 2

November 8, 2000

Review Information

The exam will cover chapters 4, 6 and sections 8.4, B.1. In addition to being able to do the computations from these sections, you should know key definitions, the statements of the main theorems, and why they are true.

Chapter 4:

Vector spaces (definition and examples)

subspaces (how to check for, examples)

linear combinations (definition)

linear independence/dependence (definition, how to check for)

span (definition, how to find good basis for)

basis (definition, how to extract a basis from a spanning set, good bases, how to create a basis from a set which does not yet span

dimension (definition, why does each basis have the same number of elements?)

solution space of homogeneous systems, i.e. matrix nullspace (definition, how to find basis of)

the four fundamental subspaces of a matrix (there were three until section 4.9) (what are they, how do you find them)

rank and nullity (definitions, theorem about)

coordinates and change of basis (definition of coordinates, how to change bases for subspaces of R^n)

orthogonal and orthonormal bases (definition, how to create via Gram-Schmidt)

projection onto subspaces of R^n (via orthogonal bases or not; Method 1 and 2)

8.4: Method of least squares

least squares solutions to inconsistent matrix equations (geometric meaning, method of computing)

application of above method to linear or quadratic data fitting.

B.1 Inner Product spaces: not on exam

abstract dot products lead to ways of measuring distance, angle and projection in vector spaces other than R^n.

Chapter 6:

linear transformations (definition, examples)

kernel and range (definition, how to find)

matrix of a linear transformation (definition, how to find, how to use)

effect of basis changes on the matrix of a linear transformation.

Practice Exam

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 100 points possible, and the point values for each problem are indicated in the right-hand margin. Good Luck!

1a) Show that the following matrix equation has no solution

(5 points)

1b) Find the least-squares solution to the problem in part (a).

(10 points)

2) Let W be the span of the vectors {[1,-1,0],[2,0,1]} in 3-space.

2a) Find an orthonormal basis for the plane W.

(10 points)

2b) Let v=[2,0,4]. Find the projection of v onto the subspace W, using your answer from (2 a)

(10 points)

2c) If you did your computations correctly, your answer to part (2b) should equal the matrix from number (1) multiplied by your least squares [x1,x2] to part (1b),

Explain why.

(5 points)

3) Let

be a matrix map from R^2 to R^2. Let S=T={[1,1],[-1,1]} be a non-standard basis for R^2. Find the matrix for L with respect to S and T. You may do this problem either of the two ways we discussed.

(20 points)

4) Let A be the four by five matrix below. Also shown are the reduced row and reduced column cecheolon forms of A.

> rref(A); #reduced row echelon form

> rcef:=M->transpose(rref(transpose(M)));

> rcef(A); #reduced column echelon form

4a) Find a ``good'' basis for the column space of A, i.e. one whose vectors have lots of zero entries.

(5 points)

4b) Prove that the vectors you found in part (a) actually are linearly independent, using the definition of linear independence.

(10 points)

4c) Find a basis for the nullspace of A.

(10 points)

4d) Find a basis for the orthogonal complement to the nullspace of A. Verify that its basis elements are orthogonal to the nullspace basis.

(5 points)

4e) Find a basis for the orthogonal complement to the column space of A

(5 points)

4f) Verify the theorem which relates rank, nullity, and domain dimension, for our matrix A above.

(5 points)