Exercise 3.2.1

Answers

1.: No. For example the rank of a $2 \times 2$ matrix with all entries $1$ has only rank $1$ .
2.: No. We have the example that the product of two nonzero matrices could be a zero matrix.
3.: Yes.
4.: Yes. This is Theorem 3.4.
5.: No. They do.
6.: Yes. This is the Corollary 2 after the Theorem 3.6.
7.: Yes. This is the argument after the definition of augmented matrix.
8.: Yes. Rank of an $m \times n$ matrix must be less than $m$ and $n$ .
9.: Yes. This means $L_{A}$ is a surjective transformation from $𝔽^{n}$ to $𝔽^{n}$ and hence a injective transformation, where $A$ is the matrix.

jephianlin

2011-06-27 00:00