
a. Let W and X both be subspaces of a vector space V. Prove that dim(WnX) > dim(W) + dim(X) - dim(V) b. Define a plane in R" (as a vector space) to be any subspace of dimension 2, and a line to be any subspace of dimension 1. Show that the intersection of any two planes in R' contains a line. c. Must the intersection of two planes in R* contain a line?
Suppose V is a finite dimensional inner product space, and dim V
= n.
If is an orthogonal subset
of V, prove that
a. W can be extended to an orthogonal basis for V.
b. is an orthogonal basis
for
c.
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) = / If(x)-g(x)| dx. 9) Let (X, di) and (Y, d2) be metric spaces. a) Prove that X × Y is a metric space with the metric b) Prove that X x Y is a metric space with the metric
7. State and prove the Law of Sines for triangles in Euclidean geometry. 8. Assume Euclidean geometry. Fix a circle and let AB and CD be two chords of the circle that intersect at point P. Prove that AP × PB = CP × PD (one both sides of the equation you are multiplying the lengths)
7. State and prove the Law of Sines for triangles in Euclidean geometry. 8. Assume Euclidean geometry. Fix a circle and let AB and...
3. Prove that every subspace S of a finitely generated subspace T of a vector space V is finitely generated, and that dim S s dim T, with equality if and only if S = T.
8. Prove that if v is a normalized (with respect to the Euclidean norm) eigenvector associated with an eigenvalue λ of a matrix A, then UTAu en v* Av
8. Prove that if v is a normalized (with respect to the Euclidean norm) eigenvector associated with an eigenvalue λ of a matrix A, then UTAu en v* Av
Let (X, 11. I be a normed vector space and let E C X be an n-dimensional subspace. (a) Prove that E is complete. (b) Prove that E is closed. (c) Prove that dim E* = n, where E* is the algebraic dual of E (the space of all linear functionals on E).
Prove in Euclidean Geometry RS^2=RA*RT
S T
. Let A be an n × n matrix. Prove that dim(span({In, A, A2,...})) ≤ n.
6.6.3 Referring to Definition 6.3, prove that (a) A totally bounded metric space is bounded (b) Show by example that there exist bounded metric spaces that are not totally bounded. (c) Consider R" with the Euclidean metric da. Show that a sust ACRis bounded if and only if it is totally bounded.