2009MT2

2009MT2 - Second Midterm MAT26500 - 11,12 Thursday November...

Info iconThis preview shows pages 1–3. 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: Second Midterm MAT26500 - 11,12 Thursday November 12, 2009 Answer as many of the following problems as possible. No tools other than a pen/pencil are permitted. You have the entire class period to work. Show all work prominently , highlighting key conclusions and explicitly answering the problem stated. There is a maximum score of 100 points. Problem 1 10 points For each of the following vector spaces and sets of vec- tors, answer the following questions: 1. Does the set span the vector space V ? 2. Is the set linearly independent? 3. Is the set a basis? Calculate, or give appropriate reason (I will assume you know the dimensions of the vector spaces given). R 2 : braceleftbiggbracketleftbigg 1 7 bracketrightbigg , bracketleftbigg 2 14 bracketrightbigg , bracketleftbigg 1 1 bracketrightbiggbracerightbigg R 3 : 1 7 1 , 1 2 2 R 4 : 1 2 , 1 1 1 , 1 3 1 1 R 3 : 1 1 , 1 1 , 1 1 P 2 [ x ] : braceleftbig t + 1 ,t- 1 ,- t 2 + t + 1 bracerightbig Answer: 1. Spans, not linearly independent, therefore not a basis. 2. Doesnt span (therefore not a basis), linearly independent. 3. Doesnt span, not linearly independent, therefore not a basis. 4. Spans, linearly independent, therefore a basis. 5. Spans, linearly independent, therefore a basis. (It is easy to write 1 ,t,t 2 as linear combinations of these vectors, therefore they span, and since P 2 [ x ] has dimension 3, it is also linearly independent and therefore a basis). 1 Problem 2 25 points Consider the matrices A = 1 3 14- 2 14- 2- 5- 25 7- 33 5 8 49- 30 103- 2- 3- 19 9- 35 , B = 1 5 7 1 3 1 1- 2 Assume in the following questions that the matrix B is obtained from the matrix A by row reduction (because it is!). 1. Find bases for null( A ) , row( A ) , col( A ) . Ans: For null( A ) , row( A ) use the matrix B to find the bases: we find for the null space the basis { [- 7- 1 2 1] T , [- 5- 3 1 0] T } for the row space the basis (pardon the bad spacing; these are just the non-zero rows of B ): [1 , , 5 , , 7] , [0 , 1 , 3 , , 1] , [0 , , , 1 ,- 2] for the column space, first observe that the 1st, 2nd, 4th columns are the columns of B containing the pivots (leading 1 s), so take the 1 , 2 , 4 th columns of A as the basis: [1 ,- 2 , 5 ,- 2] T , [3 ,- 5 , 8 ,- 3] T , [- 2 , 7 ,- 30 , 9] T 2. What does the dimension theorem claim about the matrix A . Use your answer in the previous part to verify....
View Full Document

Page1 / 9

2009MT2 - Second Midterm MAT26500 - 11,12 Thursday November...

This preview shows document pages 1 - 3. Sign up to view the full document.

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