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Exercise 6 (6.4.35, p.452) Let A e Cnxn, and let S be a k-dimensional subspace of...
of A are indexed (7) Let A A" E Cnxn and suppose that the eigenvalues so that λǐ Az < . . .-An. Show that -00 xesi vhsre S denotes the set of k-dimensional subspaces of C of A are indexed (7) Let A A" E Cnxn and suppose that the eigenvalues so that λǐ Az
Exercise 25. Let , be an orthonormal basis of a two-dimensional subspace S of R" and A xyT + (i) Show that x+y and x -y are eigenvectors of A. What are their corresponding eigenvalues? (ii) Show that 0 is an eigenvalue of R" with n - 2 linearly independent eigenvectors. (iii) Explain why A is diagonalizable. Exercise 25. Let , be an orthonormal basis of a two-dimensional subspace S of R" and A xyT + (i) Show that x+y...
please help me,thanks! 3. Let Fo be a field with 9 elements. Consider the set S () e Fo] deg(f()) 18, f( f(1) (2)) (4) 0 and (a) Compute IS. (b) Prove that S is a vector space over F (c) Compute dimF, S Let V be a vector space over F. Prove that X C V is a subspace if and only if v, w E X implies av+wEX for every aEF 3. Let Fo be a field with...
7.3 (Eigenvalues II) Let V be a vector space over K and let f,g E End(V). Show that: a) If-1 is an eigenvalue of ff, then 1 is an eigenvalue of f3. b) If u is an eigenvector off o g to the eigenvalue λ such that g(v) 0, then g(v) is an eigenvector of g o f. If, in addition, dim V < oo,then f o g and go f have the same eigenvalues c) If {ul, unt is...
(5) Prove or give a countcrcxample: If A, B E Cnx"are sclf-adjoint, then AB is also self-adjoint. (6) Let V be a finitc-dimensional inner product space over C, and suppose that T E C(V) has the property that T*--T (such a map is called a skew Hermitian operator (a) Show that the operator iT E (V) is self-adjoint (i.c. Hermitian) (b) Prove that T has purely imaginary eigenvalues (i.e. λ ίμ for μ E R). (c) Prove that T has...
Let V be the subspace of "vectors" in Hamilton's sense, that is, quat ernions with zero real part. Given a nonzero quaternion q, show that the mapping T V V defined by T(v) is an orthogonal mapping. This means that T(v). T(w) = u·w for all vectors u, w E V (again, V = the purely imaginary quat ernions) What is the mapping when q is an imaginary unit? Give its matrix for the basis i,j,k. For any nonzero quaternion...
help with p.1.13 please. thank you! Group Name LAUSD Health N Vector Spaces P.1.9 Let V be an F-vector space, let wi, W2,...,W, EV, and suppose that at least one w; is nonzero. Explain why span{w1, W2,...,w,} = span{w; : i = 1,2,..., and W; 0). P.1.10 Review Example 1.4.8. Prove that U = {p EP3 : p(0) = 0) is a subspace of P3 and show that U = span{z.z.z). P.1.11 State the converse of Theorem 1.6.3. Is it...
Please help for Question 10A.1 MATH 270 SPRING 2019 HOMEWORK 10 10A. 1. Let S be the subspace in R3 spanned by21.Find a basis for S 2. Using as the inner product (5) ( p. 246) in section 5.4 for Ps where x10, x2 -1, x3 - 2: Find the angle between p (x) = x-3 and q(x) = x2 + x + 2. b. Fnd the vector projection of p(x) on q(x) In Cl-π, π} using as an inner...
Question 1 (10 Marks) This question consists of 10 true false ansers. In cach ease, answer true if the statement is always true and false otherise. If a statement is false, 1. The set rER0 isa group under the binary operation o defined ad-be is a group under matrix addition. 3. Tho sot eRzs not an Abelian group under the binary erplain why. There is no need to show working for true statements. by a ob vab. 2. The set...
Q2. Let u and v be non-parallel vectors in Rn and define Suv (a) Does the point r lie on the straight line through q with direction vector p? (b) Does the point s lie on the straight line through q with direction vector p? (c) Prove that the vectors s and p -r are parallel. (d) Find the intersection point of the line {q+λ p | λ E R} and the line through the points u and v. Q3....