Raising the bar

Geometry of linear equations: questions for repetition

This section evolves around the concepts of linearity, linear subspaces and orthogonality. As usual, you are expected to produce at least those proofs that I give.

1. Prove linearity of the mapping generated by a matrix. This fundamental fact will have many implications. Do you think any linear mapping is generated by some matrix? Do you think a mapping inverse to a linear mapping should be linear?

2. What is the first characterization of a matrix image? In other sources it is called a column space. Later on it will be linked to the notions of linear independence and rank.

3. How would you write $latex Ax$ ($latex A$ is a matrix and $latex x$ is a vector), if $latex A$ is partitioned a) into rows and b) into columns?

4. Show that a span of some vectors is a linear subspace. Note that one- and two-dimensional examples of linear subspaces we considered are special cases of subspaces described by linear systems of equations.

5. Can spans of two different systems of vectors coincide? Give an example.

6. Show that the image and null space of a matrix are linear subspaces. This facts indicate that the subspace notion is an adequate tool for problems at hand.

7. What is the description of the set of solutions of $latex Ax=y$? It is pretty general. For example, the structure of solutions of an ordinary linear differential equation is the same.

8. Illustrate geometrically the theorem on second orthocomplement and explain why it holds.

9. What is the relationship between $latex (Ax)\cdot y$ and $latex x\cdot (A^Ty)$?

10. Derive the second characterization of matrix image.

11. What are the geometric conditions for the solvability and uniqueness of solutions of an inhomogeneous equation?

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