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Mathematics LibreTexts

14.7: Review Problems

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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=1adbc(λ1adλ2bc(λ1λ2)ab(λ1λ2)cdλ1bc+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:=vuvuuu 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

fg:=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)|nN?

What do you suspect about the vector space spansin(ax)|aR?

7.

  1. Show that if Q is an orthogonal n×n matrix then uv=(Qu)(Qv), for any u,vRn. That is, Q preserves the inner product.
  2. Does Q preserve the outer product?
  3. 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?
  4. 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),(111),(111)}. 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 = (102120122).\]

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 An=Sn? Is Mnn=SnAn?

14. The vector space V=spansin(t),sin(2t),sin(3t) has an inner product: fg:=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


This page titled 14.7: Review Problems is shared under a not declared license and was authored, remixed, and/or curated by David Cherney, Tom Denton, & Andrew Waldron.

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