Math 260, Fall 2015, Assignment 9
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]
- $S_{\mathcal{B}\rightarrow\mathcal{A}}$, the "change-of-basis matrix" that converts $\mathcal{B}$-coordinates into $\mathcal{A}$-coordinates.
- $S_{\mathcal{A}\rightarrow\mathcal{B}}$.
- Matrix of a linear transformation (with respect to an arbitrary ordered basis $\mathcal{B}$).
Carefully state the following theorems (you need not prove them):[edit]
- Transformation law for vectors (i.e. formula relating the $\mathcal{B}$-coordinates of a vector $\vec{v}$ to the $\mathcal{A}$-coordinates of $\vec{v}$; note that $\vec{v}$ itself is not transformed, only its coordinate representation).
- Transformation law for linear transformations (i.e. formula relating $[T]_{\mathcal{B}}$ to $[T]_{\mathcal{A}}$).
Solve the following problems:[edit]
- Section 3.4, problems 1, 3, 7, 17, 18, 19, 21, 25, 37, and 38.
