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