Math 260, Fall 2019, Assignment 6

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

Read:

1. Section 2.4.

Carefully define the following terms, then give one example and one non-example of each:

1. $f\circ g$ (the composition of the transformations $g:\mathbb{R}^m\rightarrow\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}^p$.
2. Inverse (of a transformation $f$).
3. Inverse (of a matrix $A$).
4. Linear subspace (of $\mathbb{R}^n$).

Carefully describe the following algorithms:

1. Procedure to determine whether a matrix $A$ is invertible, and to compute its inverse when it exists.

Carefully state the following theorems (you do not need to prove them):

1. Theorem relating matrix multiplication to composition of functions.
2. Theorem describing $f\circ f^{-1}$ and $f^{-1}\circ f$.
3. Theorem describing $AA^{-1}$ and $A^{-1}A$.
4. Formula for the inverse of a $2\times 2$ matrix (when it exists).
5. Description of all linear subspaces of $\mathbb{R}^2$.
6. Description of all linear subspaces of $\mathbb{R}^3$.

Solve the following problems:

1. Section 2.4, problems 1, 3, 5, 7, 9, 11, 17, 19, 21, and 23.
2. Suppose $A$ is an invertible $n\times n$ matrix. What can you say about its rank?
3. Suppose $A$ is a non-invertible $n\times n$ matrix. What can you say about its rank?

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