Due Oct 14 @ 11:59pm
Submit a PDF (scanned/photographed from handwritten solutions, or converted from RMarkdown or Jupyter Notebook) to Gradescope in BruinLearn.
BV exercises: 6.22, 7.13, 8.4, 10.9, 10.11, 10.19, 10.25, 10.35, 10.36, 10.42, 10.44
Q1. Prove the following facts about vector spaces.
If $\mathcal{S}_1$ and $\mathcal{S}_2$ are two vector spaces of same order, then their intersection $\mathcal{S}_1 \cap \mathcal{S}_2$ is a vector space.
If $\mathcal{S}_1$ and $\mathcal{S}_2$ are two vector spaces of same order, then their union $\mathcal{S}_1 \cup \mathcal{S}_2$ is not necessarily a vector space.
The span of a set of $\mathbf{x}_1,\ldots,\mathbf{x}_k \in \mathbb{R}^n$, defined as the set of all possible linear combinations of $\mathbf{x}_i$s $$ \text{span} \{\mathbf{x}_1,\ldots,\mathbf{x}_k\} = \left\{\sum_{i=1}^k \alpha_i \mathbf{x}_i: \alpha_i \in \mathbb{R} \right\}, $$ is a vector space in $\mathbb{R}^n$.