F09_hmwk1 - n . Problem 3. Consider the matrix, A = 3 1-1 1...

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University of California, Santa Barbara ECE 210a, Math 206a, CMPSC 211a, ME 210a, ChE 211a Matrix Analysis and Computation Fall 2009 Homework 1. Problem 1. Find a basis for the following vector spaces: 1. The set of all skew symmetric real matrices, M ∈ R 3 × 3 . Recall that a skew symmetric real matrix is one that satis±es the condition: M T = - M . 2. The set of all solutions, x ∈ R 3 , to the equation, Ax = 0, where, A = b 2 1 - 2 0 2 2 B . Problem 2. Prove that the following are subspaces or demonstrate that they are not: a) The set of all twice di²erentiable continuous functions on [0,1] is a vector space. Consider solutions x ( t ), on t [0 , 1], to the di²erential equation, t d 2 x ( t ) dt 2 + α e - t x ( t ) = 0 , α > 0 . b) The set of all singular matrices in R 2 × 2 . c) The set of all square matrices that commute with a matrix, A ∈ R n ×
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Unformatted text preview: n . Problem 3. Consider the matrix, A = 3 1-1 1 9 3 5-6-4 . Find a basis for each of the subspaces: R ( A ), R ( A T ) , N ( A ) and N ( A T ) . What is the rank of A ? Problem 4 Calculate the matrix coordinate representation for the linear operator, D ( p ( t )) = dp ( t ) dt-p ( t ) , on the space of polynomials of degree three on the interval [-1,1]. Use Chebyshev polynomials (of the ±rst kind) as the basis for the space. If you are unfamiliar with Chebyshev polynomials you can ±nd a recurrence relation for generating them on Wikipedia. Problem 5. Consider Ax = b and suppose we are given a vector c ∈ R ( A T ) . Show that c T x is constant for all x that are solutions to the normal equations A T Ax = A T b . 1...
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This note was uploaded on 12/10/2009 for the course ECE 210a taught by Professor Roysmith during the Fall '09 term at Santa Barbara City.

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