MATH 3210 Class Notes 01-27-2025

notevector-spaces

Proposition

A subset $U$ of $V$ is a subspace of $V$ if and only if $U$ satisfies:

1. $0\in U$
2. $u,w\in U$ implies $u+w\in U$.
3. $a\in \mathbb{F}$, $u\in U$ implies $au\in U$.

Proof

"$⟹$" Assume $U$ subspace of $V$. Then $U$ is a vector space with $0\in U$. Therefore, $U$ trivially satisfies conditions 1), 2), and 3).

"$⟸$" Assume $U\subseteq V$ satisfies conditions 1), 2), and 3). Clearly, $0\in U$ (by assumption), and addition and scalar multiplication are well-defined on $U$. If $u\in U$, then, by 3), $(-1)u=-u\in U$. So $U$ contains all additive inverses. Since $U\subseteq V$, any $u\in U$ is also in $V$. So the vector space properties (associativity, distributivity, commutativity, and existence of an identity) are trivially held for any elements in $U$ since they are held for those elements in the vector space $V$.

Example

1. For which $b\in \mathbb{F}$ is $U=\{(x_1,x_2,x_3)\in {\mathbb{F}}^3\mid x_1=5x_3+b\}$ a subspace of ${\mathbb{F}}^3$? Since $0\in U$, we must have $0=5(0)+b$ so $b=0$.
2. The set of differentiable real-valued functions on $\mathbb{R}$ is a subspace of ${\mathbb{R}}^{\mathbb{R}}$:
   1. $0\in {\mathbb{R}}^{\mathbb{R}}$ and $0$ is differentiable.
   2. If $f,g$ are differentiable, so is $f+g$.
   3. If $f$ is differentiable and $\lambda \in \mathbb{R}$, then $\lambda f$ is differentiable. $⟹$ differentiable subspace of ${\mathbb{R}}^{\mathbb{R}}$. Similarly, continuous functions are a subspace of ${\mathbb{R}}^{[0,1]}$.
3. The set of complex sequences with ${\lim }_{k\to \infty }{x}_k=0$ is a subspace of ${\mathbb{C}}^{\infty }$.

Sums of Subspaces

Definition

Suppose $V_1,V_2,\dots ,V_m$ are subspaces of $V$. The sum of $V_1,\dots ,V_m$ is

$$V_1+\dots +V_m=\{v_1+\dots +v_m\mid v_k\in V_k\forall k=1,\dots ,m\}.$$

Example

1. Let $U=\{(x,0,0)\in {\mathbb{F}}^3\mid x\in \mathbb{F}\}$ and $W=\{(0,y,0)\in {\mathbb{F}}^3\mid y\in \mathbb{F}\}$, then $U+W=\{(x,y,0)\in {\mathbb{F}}^3\mid x,y\in \mathbb{F}\}$.
2. Let $U=\{(x,x,y,y)\in {\mathbb{F}}^4\mid x,y\in \mathbb{F}\}$ and let $W=\{(x,x,x,y)\in {\mathbb{F}}^4\mid x,y\in \mathbb{F}\}$. Then we claim $U+W=\{(x,x,y,z)\in {\mathbb{F}}^4\mid x,y,z\in \mathbb{F}\}$.

Proposition

$V_1+\cdots +V_m$ is the smallest subspace of $V$ containing $V_1,V_2,\dots ,V_m$.

Defined Direct Sums

We know $w\in V_1+\cdots +V_m$ can be expressed as $w=v_1+\cdots +v_m$.