math 304 practice final exam

math 304 practice final exam - MATH 304505 Sample problems...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 304505 Sample problems for the final exam Spring 2011 Any problem may be altered or replaced by a different one! Problem 1 (15 pts.) Find a quadratic polynomial p(x) such that p(-1) = p(3) = 6 and p (2) = p(1). Problem 2 (20 pts.) Let v1 = (1, 1, 1), v2 = (1, 1, 0), and v3 = (1, 0, 1). Let L : R3 R3 be a linear operator on R3 such that L(v1 ) = v2 , L(v2 ) = v3 , L(v3 ) = v1 . (i) Show that the vectors v1 , v2 , v3 form a basis for R3 . (ii) Find the matrix of the operator L relative to the basis v1 , v2 , v3 . (iii) Find the matrix of the operator L relative to the standard basis. 1 1 Problem 3 (20 pts.) Let A = 0 2 1 1 1 3 0 0 1 -1 . 0 1 0 0 (i) Evaluate the determinant of the matrix A. (ii) Find the inverse matrix A-1 . (i) Find all eigenvalues of the matrix B. (ii) Find a basis for R3 consisting of eigenvectors of B. (iii) Find an orthonormal basis for R3 consisting of eigenvectors of B. (iv) Find a diagonal matrix X and an invertible matrix U such that B = U XU -1 . 1 1 1 Problem 4 (25 pts.) Let B = 1 1 1 . 1 1 1 Problem 5 (20 pts.) Let V be a subspace of R4 spanned by vectors x1 = (1, 1, 0, 0), x2 = (2, 0, -1, 1), and x3 = (0, 1, 1, 0). (i) Find the distance from the point y = (0, 0, 0, 4) to the subspace V . (ii) Find the distance from the point y to the orthogonal complement V . 1 Bonus Problem 6 (15 pts.) (i) Find a matrix exponential exp(tC), where C = and t R. (ii) Solve a system of differential equations dx = 3x + y, dt dy = 3y dt 3 1 0 3 subject to the initial conditions x(0) = y(0) = 1. Bonus Problem 7 (15 pts.) Consider a linear operator K -4 7 1 K(x) = Dx, where D = 1 -4 9 8 4 The operator K is a rotation about an axis. (i) Find the axis of rotation. (ii) Find the angle of rotation. : R3 R3 given by 4 8 . 1 2 ...
View Full Document

Page1 / 2

math 304 practice final exam - MATH 304505 Sample problems...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online