advertisement

Homework 12 Due: Wednesday, December 10, 2008 Light homework set – start reviewing for the final! 1. Let U be a subspace of a finite-dimensional inner product space V. In class, we saw that for any v ∈ V, there is a u ∈ U and a w ∈ U ⊥ such that v = u + w. This u is just the projection of v onto U, i.e., m u= ∑ hv, ei iei , i= 1 where {e1 , · · · , em } is an orthonormal basis for U. (a) w is defined as v − u, where u has the form given above. Prove that projection onto U ⊥ is a linear transformation. (b) Suppose the basis {e1 , · · · , em } of U is not actually orthonormal. In this case, does PU (u) = u for every u ∈ U? If so, why? If not, please give a counterexample. (H INT: Keep it simple; a line in R2 makes a nice subspace....) 2. Form the degree 1, 2, and 3 least squares fits of the points (1, 4), (2, 1), (3, 3), and (4, 1), and plot all three curves and the four points on the same axes. 1 3 3. Consider the vectors v = 4, w = 3 ∈ R3 ; equip R3 with the standard inner product. 7 2 0 1 2 (a) Write v and w as linear combinations of the basis vectors 0, 2, and 1. Let [v] 1 0 1 denote the vector of coefficients from writing v in this way (and, similarly, [w] for w). (b) Form the Gram matrix G for the basis of the previous part. (c) Use G to compute the inner product hv, wi as [v] T G [w]. (d) Confirm your answer to the previous part by computing the dot product of v and w. Professor Dan Bates Colorado State University M369 Linear Algebra Fall 2008