14.7: Review Problems
( \newcommand{\kernel}{\mathrm{null}\,}\)
1. Let D=(λ100λ2)
a) Write D in terms of the vectors e1 and e2, and their transposes.
b) Suppose P=(abcd) is invertible. Show that D is similar to
M=1ad−bc(λ1ad−λ2bc−(λ1−λ2)ab(λ1−λ2)cd−λ1bc+λ2ad)
c) Suppose the vectors (a,b) and (c,d) are orthogonal. What can you say about M in this case? (Hint: think about what MT is equal to.)
2. Suppose S=v1,...,vn is an orthogonal (not orthonormal) basis for Rn. Then we can write any vector v as v=∑icivi for some constants ci. Find a formula for the constants ci in terms of v and the vectors in S.
3. Let u,v be linearly independent vectors in R3, and P=spanu,v be the plane spanned by u and v.
(a) Is the vector v⊥:=v−u⋅vu⋅uu in the plane P?
(b) What is the (cosine of the) angle between v⊥ and u⊥?
(c) How can you find a third vector perpendicular to both u and v⊥?
(d) Construct an orthonormal basis for R3 from u and v.
(e) Test your abstract formulæ starting with u=(1,2,0) and v=(0,1,1).
4. Find an orthonormal basis for R4 which includes (1,1,1,1) using the following procedure:
(a) Pick a vector perpendicular to the vector
v1=(1111)
from the solution set of the matrix equation
vT1x=0.
Pick the vector v2 obtained from the standard Gaussian elimination procedure which is the coefficient of x2.
(b) Pick a vector perpendicular to both v1 and v2 from the solutions set of the matrix equation
(vT1vT2)x=0.
Pick the vector v3 obtained from the standard Gaussian elimination procedure with x3 as the coefficient.
(c) Pick a vector perpendicular to v1,v2, and v3 from the solution set of the matrix equation
\[(vT1vT2vT3)x = 0.
Pick the vector v4 obtained from the standard Gaussian elimination procedure with x3 as the coefficient.
(d) Normalize the four vectors obtained above.
5. Use the inner product
f⋅g:=∫10f(x)g(x)dx
on the vector space V=span1,x,x2,x3 to perform the Gram-Schmidt procedure on the set of vectors 1,x,x2,x3
6. Use the inner product on the vector space V=spansin(x),sin(2x),sin(3x) to perform the Gram-Schmidt procedure on the set of vectors sin(x),sin(2x),sin(3x).
What do you suspect about the vector space spansin(nx)|n∈N?
What do you suspect about the vector space spansin(ax)|a∈R?
7.
- Show that if Q is an orthogonal n×n matrix then u⋅v=(Qu)⋅(Qv), for any u,v∈Rn. That is, Q preserves the inner product.
- Does Q preserve the outer product?
- If u1,...,un is an orthonormal set and λ1,···,λn is a set of numbers then what are the eigenvalues and eigenvectors of the matrix M=∑ni=1λiuiuTi?
- How does Q change this matrix? How do the eigenvectors and eigenvalues change?
8. Carefully write out the Gram-Schmidt procedure for the set of vectors {(111),(1−11),(11−1)}. Are you free to rescale the second vector obtained in the procedure to a vector with integer components?
9.
a) Suppose u and v are linearly independent. Show that u and v⊥ are also linearly independent. Explain why u,v⊥ is a basis for spanu,v.
b) Repeat the previous problem, but with three independent vectors u,v,w.
10. Find the QR factorization of $$M = (102−120−1−22).\]
11. Given any three vectors u,v,w, when do v⊥ or w⊥ of the Gram-Schmidt procedure vanish?
12. For U a subspace of W, use the subspace theorem to check that U⊥ is a subspace of W.
13. Let Sn and An define the space of n×n symmetric and anti-symmetric matrices respectively. These are subspaces of the vector space Mnn of all n×n matrices. What is dimMnn, dimSn and dimAn? Show that Mnn=Sn+An. Is A⊥n=Sn? Is Mnn=Sn⊕An?
14. The vector space V=spansin(t),sin(2t),sin(3t) has an inner product: f⋅g:=∫2π0f(t)g(t)dt. Find the orthogonal compliment to U=spansin(t)+sin(2t) in V. Express sin(t)−sin(2t) as the sum of vectors from U and UT.
Contributor
David Cherney, Tom Denton, and Andrew Waldron (UC Davis)