8 problems.
- If \(f\circ g\) is injective then \(g\) is injective, but \(f\) need not be
- The kernel of a linear map \(\mathbb{R}^n\to\mathbb{R}^m\) is a subspace
- A spanning set with no proper spanning subset is linearly independent
- Diagonalize \(\begin{bmatrix} 1 & 2 \\ 4 & -1 \end{bmatrix}\) and compute \(A^2\)
- Continuity preserves sequential limits of \(g\)
- The sequence \(x_{n+1}=\frac14 x_n^2+1\) with \(x_1=1\) is increasing and bounded by \(2\)
- Continuous solutions of \(f(x)=5+\int_0^x 3f(t)\,dt\)
- Examples - a false converse, continuous not differentiable, separate continuity, non-Cauchy bounded sequence, and a series converging exactly on \([0,2]\)