2009bfinsol - M346 Final Exam, December 15, 2009 1) The...

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: M346 Final Exam, December 15, 2009 1) The matrix A = 1 3 2 5 2 1 1 1 1 2 3 3 3 5 7 row-reduces to B = 1 4 / 11 1 13 / 11 1 10 / 11 . a)Find all solutions to A x = 0. These are the same as the solutions to B x = 0, namely all multiples of (4 / 11 , 13 / 11 , 10 / 11 , 1) T , or equivalently all multiples of (4 , 13 , 10 , 11) T . b)Find a basis for the column space of A . Since there are pivots in the first three columns of B , the first three columns of A for a basis. That is, the answer is 1 2 3 , 3 1 1 3 , 2 1 2 5 . c)In R 3 [ t ], let V be the span of the vectors { 1 + 2 t + 3 t 3 , 3 t + t 2 + 3 t 3 , 2+ t +2 t 2 +5 t 3 , 5 t +3 t 2 +7 t 3 } . What is the dimension of V ? Find a basis for V . If you express things in coordinates with respect to the standard basis { 1 , t, t 2 , t 3 } , this becomes the same problem as (b). V is 3-dimensional, as a basis consists of those polynomials whose coordinates are the answer to (b), namely { 1+2 t +3 t 3 , 3 t + t 2 +3 t 3 , 2+ t +2 t +2+5 t 3 } . Note that the answer is NOT a matrix or a list of column vectors. Those are just the coordinates of the answer, not the answer itself. 2. a) Find the eigenvalues of 3 5 16 4 3 11 15 1 4 1 2 . You do not need to find the eigenvectors. The matrix is block triangular, with an upper left 1 1 block and a lower right 3 3 block. The 3 3 block is itself block triangular, with an upper left 2 2 piece parenleftbigg 3 11 15 1 parenrightbigg and a lower right 1 1 piece. The rows of the 2 2 piece sum to 14, and the trace is 2, so that piece has eigenvalues 14 and 12, and the whole matrix has eigenvalues 3 , 14 , 12 , 2. b) Find the eigenvalues and eigenvectors of parenleftbigg 3 8 2 3 parenrightbigg ....
View Full Document

This note was uploaded on 04/10/2011 for the course M 346 taught by Professor Radin during the Spring '08 term at University of Texas at Austin.

Page1 / 5

2009bfinsol - M346 Final Exam, December 15, 2009 1) The...

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