Friedberg, Insel, and Spence Linear algebra, 4th ed.
SOLUTIONS REFERENCE
Michael L. Baker <mbaker@lambertw.com>
UNIVERSITY OF WATERLOO
August 21, 2010
Preface
The aim of this document is to serve as a reference of problems and solutions from the fourth ed
Math 146
Instructor: L.W. Marcoux
Assignment One.
Due: Friday, January 17, 2014
Question 1. An example or counterexample of a vector space
Consider the set V = R+ = cfw_x R : x > 0 equipped with an addition (denoted by ]) defined
via x]y = xy, where xy is
Math 146
Instructor: L.W. Marcoux
Assignment Two.
Due: Friday, January 24, 2014
Question 1. Linear combinations
(a) Consider R3 as a vector space over R. Determine whether or not (2, 1, 0) can be expressed
as a linear combination of (1, 2, 3) and (1, 3, 2
Math 146
Instructor: L.W. Marcoux
Assignment Four.
Due: Friday, February 14, 2014
Question 1. Extending bases from subspaces to spaces
Let W be a subspace of a (not necessarily finite-dimensional) vector space V . Prove that any
basis for W is a subset of
Math 146
Instructor: L.Marcoux
Assignment Seven.
Due: Friday, March 14 , 2014
Question 1. An algebraic Hahn-Banach Theorem
Let 0 6= V be a vector space over a field F and let W 6= V be a subspace of V . Prove that there
exists a linear functional V such t
Math 146
Instructor: L.W. Marcoux
Assignment Six.
Due: Friday, March 7, 2014
Question 1. Linear maps take subspaces to subspaces
Let V and W be finite-dimensional vector spaces and T L(V, W ). Let Z be a subspace of V .
(a) Prove that T (Z) := cfw_T z : z
Math 146
Instructor: L.W. Marcoux
Assignment Eight
Due: Friday, March 21, 2014
Question 1. The double dual
Let V and W be finite-dimensional vector spaces and let 1 and 2 be the isomorphisms between
V and V and W and W respectively. Let T : V W be linear,
Math 146
Instructor: L.W. Marcoux
Assignment Three.
Due: Friday, January 31, 2014
Question 1. Generating sets
Do the polynomials x3 2x2 + 1, 4x2 x + 3,
answer.
and 3x 2 generate P3 (R)? Justify your
Question 2. A basis for a subspace
Let W denote the subs