Sections, concepts, and problems

1.1. Introduction. An overview of the basic operations of vectors in Euclidean space, including equations of lines and planes, which sets up the concept of vector space in the next section.
1, 2, 3, 7 (don't use components for this problem if you can help it; there is a more elegant way in the spirit of Figure 1.1)

1.2. Vector Spaces. The definition of a vector space (its eight axioms), some common examples, and a few properties of vector spaces.
1, 4, 7, 8 (list the axioms you use), 12 (you can assume Example 3), 18, 21 (note that Z = V x W)

1.3. Subspaces. Subspaces, how to show if a subspace of a vector space is a subspace, and examples.
1, 3 (start this way: [(aA + bB)^t]_ij = ...), 5 (you can use problem 4), 13, 17 (this is a handy way to show a subset is a subspace), 22 31 (cosets!)

1.4. Linear combinations and systems of linear equations. Linear combinations of vectors, systems of linear equations, the span of a set, and the notion of a spanning set for a vector space.
1, 5, 6, 10, 11

1.5. Linear dependence and linear independence. A nice section on what these terms mean.
1, 2a-e, 4, 7, 9, 10, 11 (Z₂ is the set {0,1} that has addition and multiplication given by the tables for GF(2) on Wolfram's page on finite fields), 12 (for the corollary, think contrapositive), 16 (if this seems hard, then you don't know your definition), 20 (Plug t=0 into both the equation from a linear combination of f and g, and f' and g')

1.6. Bases and dimension. The definitions of those words, results about finite bases (including the Replacement Theorem), how the dimensions of a subspace and the full space compare, and an application to the Lagrange Interpolation Formula.
1, 4, 5, 9, 12, 13, 14, 19, 20a, 26, 29a

Other Chapters

Chapter 1 | Chapter 2 | Chapter 3 | Chapter 4 | Chapter 5 | Chapter 6