About the upcoming midterm exam: click here.

Due to the midterm, there will be no assignment due this week.

Required reading: Section 2.1, 2.2

Material covered this week includes: Dimension theorem (also called Rank-Nullity Theorem), notions of one-to-one (=injective), onto (=surjective), invertible (=bijective); linear isomorphisms. The fact that the inverse of a linear isomorphism is again linear.

Additional homework (do not hand in).

• Section 2.1, Problems 1, 2, 3, 21, 24, 25, 26, 27
• Section 2.2, Problems 1, 4, 13, 16
• Let $V$ be a vector space, $W_1,W_2$ two subspaces. Prove $ \dim(W_1+W_2)+\dim(W_1\cap W_2)=\dim(W_1)+\dim(W_2).$
• Let $F$ be a field, and $X$ a finite set. Show that the set of functions $\mathcal{F}(X,F)$, with addition and scalar multiplication defined `pointwise', is a vector space, and describe a basis of this vector space. Hint: Write $X=\{c_1,\ldots,c_n\}$ and follow the strategy from the Lagrange Interpolation Theorem. More generally, if $V$ is a vector space over $F$, show that $\mathcal{F}(X,V)$ is naturally a vector space, and that a basis of $V$ determines a basis of this vector space.