MATH 3210 Class Notes 01-31-2025

notevector-spaces

Continuing with Direct Sums

Examples

2. If $V_k=\{(0,\dots ,0,x_k,0,\dots ,0)\in {\mathbb{F}}^n\}$, then ${\mathbb{F}}^n=V_1\oplus \cdots \oplus V_n.$
3. Nonexample: Let $V_1=\{(X,y,0)\in {\mathbb{F}}^3\},V_2=\{(0,0,z)\in {\mathbb{F}}^3\},V_3=\{(0,y,y)\in {\mathbb{F}}^3\}$. Question: ${\mathbb{F}}^3=V_1\oplus V_2\oplus V_3$? It is clear that ${\mathbb{F}}^3=V_1+V_2+V_3$: $(a,b,c)=(a,b,0)+(0,0,c)+(0,0,0)$. However, $(0,0,0)=(0,0,0)+(0,0,0)+(0,0,0)=(0,1,0)+(0,0,1)+(0,-1,-1).$ Thus, ${\mathbb{F}}^3$ is not a direct sum of $V_1\oplus V_2,\oplus V_3.$

Proposition

Suppose $V_1,\dots ,V_m$ are subspaces of $V$. Then, $V_1+\cdots +V_m=V_1\oplus \cdots V_m$ if and only if the only way to write $0$ as a sum of $v_1+\cdots +v_m$ is if all the $v_k$‘s are $0$.

Proof

Homework!

Proposition

Suppose $U$ and $W$ are subspace of $V$, then $U+W=U\oplus W$ if and only if $U\cap W=\{0\}$.

Proof

"$⟹$" Assume $U+W$ is a direct sum. Let $v\in U\cap W$, then $0=v+(-v)$. Since $U,W$ are vector spaces, $-v\in U\cap W$ as well. By the previous proposition, $0$ has a unique representation as a sum of elements of $U$ and $W$: $0=0+0$. Thus, $v=0$, so $U\cap W=\{0\}$.

"$⟸$" Assume $U\cap W=\{0\}$. Suppose $u\in U$ and $w\in W$, and $u+w=0$. By previous proposition, $U+W=U\oplus W$ iff $u=w=0$.

Finite Dimensional Vector Spaces

$V$ is a vector space over $\mathbb{F}$.

Defined Linear Combination, Span, Finite-Dimensionality

Discussed polynomial space as an example of a vector space.