Name:_______________________________________________

M260 EXAM2 {3.1-3.4, 5.1-5.4}

NO BOOKS, NO NOTES, NO CALCULATORS

SHOW ALL WORK

TIME: 75 MINUTES

EACH PROBLEM IS WORTH 10 POINTS FOR A TOTAL OF 60 POINTS.

1. Let . Find a basis for the kernels and Images of A and and give a description of them as subspaces of . (Hint: check = ? and = ?)

2. Suppose A is any 3 x 3 matrix.

a. What is the relationship between the kernels of A and ? Prove your statement. (Hint: Is one included in the other? i.e. If x ker(A) then ....)

b. What is the relationship between the Images of A and ? Prove your statement. (Hint: if x Im() then ....)

3. Find a basis of the ker(A) where then show that your basis elements are in the kernel.

4. Find the orthogonal projection of onto the subspace V of spanned by

Thus find . (Hint: you need an orthonormal basis of V.)

5. a. What is the definition of orthogonal?

b. If are both orthogonal, Prove that TL (the composite) is also orthogonal.

c. State Pythagoras’s theorem? (Note: “If, and only if”)

d. Show that if is orthogonal and if u, v are orthogonal in then so are T(u), T(v). (Hint: use Pythagoras’s theorem)

6. (1) Let be vectors in . When do these vectors form an orthonormal basis? (i.e. give the definition.)

(2) By using the Gram-Schmidt process, find an orthonormal basis of the subspace V of spanned by the vectors , and .