Linear Algebra Math 310
Spring 2012
Assignment Sheet 3

This sheet specifies a selection of exercises from the text, and in some cases a few additional exercises. The notation 1-33o means the odd problems between 1 and 33, inclusive. 

Section 4.1 1,3, 4, 5-21o,(27, 29-33)*  PLUS (complete the handout on unusual vector spaces.)*

Section 4.2   1-25o,27*,29*,30*,31,33,35*, plus:

  1. Without looking in text, define Null Space, Column Space, Range, and Kernel
  2. Makeup a definition of the Row Space of a matrix in analogy with the definition of column space. Show that every for every v in the row space of A and for every w in the null space of A, the dot product of v and w is 0.

Section 4.3   1-13o,14,15,19,(21-26,29-32)*,plus

  1. Without looking in text, define linear independence, basis, standard basis.
  2. Can a set of 6 or more vectors in R5 be linearly independent? Why? Can a basis for R5 have 6 or more elements? Why?
  3. Can a set of 4 or fewer vectors in R5 be spanning set for R5? Why? Can a basis for R5 have 4 or fewer elements? Why?

Section 4.4   1-17o,18,(19-26)*,27,29

Section 4.5   1-27o,(29-32)*

Section 4.6   1-25o,27-30

Section 4.7   1-13o, plus

  1. The set {1,x,x2,x3,x4 } is a basis for the polynomials of degree less than or equal to 4. An alternate basis is given by the translated polynomials {1, x-1, (x-1)2, (x-1)3, (x-1)4 }. Find the change of coordinates matrix which transforms coordinate vectors relative to the first basis into coordinate vectors relative to the second basis.
  2. If you graph the polynomials in the first basis and those in the second, how are the two sets of graphs related?
  3. Find the inverse matrix for the answer to part 1.
  4. Consider a third basis for the polynomial space: {1, x+1,(x+1)2, (x+1)3, (x+1)4 }. Find the change of coordinates matrix going from the first basis {1, x, x2, x3, x4 } to this new basis.
  5. Compare the answers to the last two questions. How do you explain what you observe?