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Problems:

You should work all of the WQS homework problem over chapter 3 and chapter 4. Here are 15 review problems for you to work through. Some of the questions (e.g. 1 def and 11 e ) require more than might be expected on the exam, but working on them will help prepare you.

1. Suppose a 10 pound mixture of p, c, w, and h pounds respectively of peanuts, cashews, walnuts, and halzelnuts. The cost of the mixture is to be 7 dollars where the 4 nuts are respectively 3/10, 1, 4/5, and 1/2 dollars per pound. So any mixture [p,c,w,h] must satisfy the system of 2 equations and . Note: there are mixtures x = [p,c,w,h] with some negative components which satisfy the equations. In parts d, e, and f below, we restrict our attention to 'real' mixtures, ones with no negative component.

a) Write a matrix equation Ax = b that descibes the system.

b) What is the dimension of the null space of A? Find a basis for the null space of A.

c) Find a particular solution (mixture) to the equation and then describe the complete solution to Ax = b.

d) What real mixture will have the most peanuts? the most cashews?

e) What real mixture will have the most walnuts? the most hazelnuts?

f) Describe all real mixtures.

Solution:

2. What is the rank of the 4 by 4 checkerboard matrix A= [Maple Math] ? Can you make the rank 4 by changing one entry in A? Explain your answer.

Solution

3. Find a basis for the column space of [Maple Math] . (n is an unspecified constant)

Solution

4. Here are 3 vectors in : [Maple Math] . and . Find a matrix equation Ax = b with complete solution . Hint: Take A to be the null space of a certain 2 by 4 matrix.

Solution

5. In problem 4, show that there are infinitely many different equations Ax = b, i.e. infinitely many choices of the matrix , having the given complete solution.

Solution

6. Suppose V is a vector space with basis , and . Let . Show that and u form a basis for V.

Solution

7. Suppose u, v, w, and z are linearly independent vectors in V. Show that u, v, and z are linearly independent.

Solution

8. You have 5 vectors in . Let V be the span of these vectors. Can you say what the dimension of V must be? Why?

In the exam, you may expect some concrete vectors given to you and you may go thru the null space calculations as usual. You should try some concrete examples.

Solution

9. Suppose V is a vector space with basis , and . Let , and . Show that , and are linearly dependent by exhibiting a nontrivial solution to .

Solution

10. Suppose we have 4 vectors in a 3 dimensional space. We know from a theorem that these vectors must be linearly dependent. Prove this without using that theorem. (You may use the fact that if A is a 3 by 4 matrix, then the equation Ax = O has a nonzero solution.)

Solution

11. Note 11e requires some ideas from other math courses!
a) Find a basis for the plane .

b) Write down a matrix A whose column space is the plane in a).

c) Use A to form the matrix which projects onto the plane in a).

d) The segment from [-1,0,0] to [1,0,0] on the x-axis projects to a segment on the plane in a). Find the length of that segment.

e) The projection of the unit circle in the xy plane onto the plane in a) is an ellipse. Find the length of the major axis of the ellipse.

Solution

12. Find the least squares solution to the over determined system: [Maple Math]

Solution

13. Will the orthogonal complement to the plane spanned by the vectors [Maple Math] and in be a 1, 2 or 3 dimensional subspace of ? Find a basis for it. (Hint: find the null space of a certain 2 by 4 matrix).

Solution

14. Find all quadratic polynomials [Maple Math] which pass through the points [0,1] and [1,2]. Is there only one or is there a 'line' of them or is there a 'plane' of them?

Solution

15. Find the quadratic polynomial [Maple Math] which most nearly passes through [0,1], [1,1], [2,0], and [3,1] in the sense of least squares.

Solution