Math 260, Fall 2017, Assignment 8
From cartan.math.umb.edu
Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.
- - John Locke, Second Reply to the Bishop of Worcester
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Coordinates (of a vector $\vec{x}$ with respect to a basis $\mathcal{B}=\{\vec{v}_1,\dots,\vec{v_k}\}$).
Carefully state the following theorems (you need not prove them):[edit]
- Theorem relating the size of a linearly independent set in a subspace $S$ to that of a spanning set in the same subspace.
- Invariance of dimension.
- Theorem concerning the existence of bases.
- Theorem concerning the extension of linearly independent sets to bases.
- Theorem concerning the refinement of spanning sets to bases.
- Theorem relating the dimensions of "nested" subspaces $S\subseteq T$.
- Theorem concerning nested subspaces $S\subseteq T$ whose dimensions are equal.
Solve the following problems:[edit]
- Section 3.3, problem 27, 34, 35, 38, and 64.