8 problems.
- Adjoining a vector already in a span does not change the span
- No \(3\times 2\) matrix satisfies \(A^t A=I_2\) and \(AA^t=I_3\)
- The formula \(\sum_{i=1}^n i^2 = n(n+1)(2n+1)/6\)
- \(x > \ln x\) for all \(x > 0\)
- Every linear map \(\mathbb{R}^2\to\mathbb{R}^2\) is continuous
- No divergent series of positive terms has convergent series of square roots
- Differentiability in \(\mathbb{R}^n\), and \(xy/(x^2+y^2)\) is not differentiable at the origin
- Line integral of \(2x\,dx+x^2y\,dy\) on the unit square