Let A be a 3 x 3 orthogonal matrix and let |A| = 1. Show L = 1 is an eigenvalue of A. Let W be the eigenspace of A corresponding to L = 1. Show dim(W) = 3 if A = I and dim(W) = 1 if A is not equal I. |
|
0023 |
Let A be an n x n matrix and let l0 be a root (not necessarily real) of the characteristic equation of A. Show l0k is a root of the characteristic equation of Ak. Let A be a 100 x 100 matrix such that A2 = I and let t = trace(A). Show t is an even integer and -100 =< t =< 100. |
|
0022 |
Without finding the characteristic equation find an eigenvalue of matrix A. |
|
0021 |
Let F(x, y, z) = y3 + x2 - 3y2 + z2 + xz + 3x. Find the stationary points of F. Examine if F has a local extrema or a saddle point at the points found in (a). |
|
0020 |
Consider the curve: x2 + y2 + 2xy + 3x + y - 1 = 0. Find a suitable coordinate system x_, y_ to put the above conic section in standard form. Find the x, y coordinates of the origin of the final coordinate system x_, y_. Find the x_, y_ coordinates of the point (x, y) = (1, -1). |
|
0019 |
Let B be an n x n invertible matrix and let A = BTB. Show A is positive definite. Let A be an n x n (symmetric) positive definite matrix. Show there exists an invertible matrix B such that BTB = A. |
|
0018 |
Let m be the minimum and M the maximum value of f(x, y, z) = 2x2 + 2y2 + z2 - 2xz + 2yz subject to x2 + y2 + z2 = 1. Without using calculus, find m and M. Find a point (x0, y0, z0) on the sphere x2 + y2 + z2 = 1 such that f(that point) = m, and f(that point) = M. |
|
0017 |
Solve an initial value problem for a system of differential equations. |
|
0016 |
Given 2 x 2 matrix A. Find An. Find lim as n -> infinity of 1/(23n-1)An. Find all 2 x 2 matrices B such that B2 = A. |
|
0015 |
Let E = integral from -1 to 1 of (ex - p(x))2 with respect to x where p(x) is an element of P2. Find p(x) so that E is minimum. Find the exact minimum value of E. |
|
0014 |
Let A(0,3) B(1,2) C(2,4) D(3,4). Find the exact least squares straight line to fit the points. Find the exact least squares quadratic to fit the points. |
|
0013 |
Find the least squares solution to a system of linear equations. |
|
0012 |
Let Rn have the Euclidean inner product and let u be a unit vector in Rn. Let H = In - 2uuT. Show that H is both symmetric and orthogonal and hence its own inverse. |
|
0011 |
Let A be an n x n matrix such that A5=0-matrix. Let B1 = I - A and B2 = I - A2. Show B1 and B2 are invertible. Express B1-1 and B2-1 in terms of A. |
|
0010 |
Let W be the subspace of R5 spanned by v1, v2, v3, and v4. Find a homogeneous system where solution space is W. |
|
0009 |
Let R4 have the Euclidean inner product and let W be the subspace of R4 spanned by v1, v2, v3 (defined). Find an orthonormal basis for W. Find an orthonormal basis for W-perp. Find two vectors when summed equal a third given vector where one vector is in W and the other is in W-perp. |
|
0008 |
Let V be an inner product space, and let u, v, w be elements of V with given norms and inner products, find angle between v and w, compute an inner product, and find a distance. |
|
0007 |
Express rank(A) as function of parameter t. |
|
0006 |
Given an inner product defined in R2, find the angle between 2 vectors, the distance between 2 vectors, a unit vector orthogonal to a vector, and an orthonormal basis for the space. |
|
0005 |
Let S={v1, v2, v3} be basis for V and S*={u1, u2, u3} where u1, u2, and u3 are linear combinations of v1, v2, and v3, show S* is a basis for V and find (v2) in S*. |
|
0004 |
Show the set of polynomials p(x) of degree n<=2 satisfying p(-1)=0 is a subspace of P2. |
|
0003 |
Given a set of four 4-element vectors, find a subset of the set that forms a basis for the space (a subspace of R4) spanned by the orignal set. |
|
0002 |
Show that either the row or column vectors of any rectangular matrix are linearly dependent. |
|
0001 |
Find bases for the row space, column space, and null space of a matrix. |
|
0000 |