Page 313 - 35Linear Algebra

17.3 Review Problems 313

T
T
• v M Mv ≥ 0. In case you are concerned (you don’t need to be)
and for future reference, the notation v ≥ 0 means each component
i
v ≥ 0.
T
T
• If v M Mv = 0, then v = 0.
n
(Hint: Think about the dot product in R .)

Hint

3. Rewrite the Gram-Schmidt algorithm in terms of projection matrices.

4. Show that if v 1 , . . . , v k are linearly independent that the matrix M =
T
(v 1 · · · v k ) is not necessarily invertible but the matrix M M is invert-
ible.

5. Write out the singular value decomposition theorem of a 3 × 1, a 3 × 2,
and a 3×3 symmetric matrix. Make it so that none of the components
of your matrices are zero but your computations are simple. Explain
why you choose the matrices you choose.

6. Find the best polynomial approximation to a solution to the differential
2
equation d f = x + x by considering the derivative to have domain
dx
2
and codomain span {1, x, x }.
(Hint: Begin by defining bases for the domain and codomain.)

313