> [!NOTE] Definition (vector subspace) > Let $V$ be a [[Vector Space|vector space]]. A *subspace* of $V$ is a non-empty $W\subset V$ with the property that for any $v,w\in W$ and $\lambda v\in W$, we also have $v+w\in W$ and $\lambda v \in W$. > [!NOTE] Definition (non-trivial subspace of $\mathbb{F}^{n}$) > The *trivial* subspaces are $\{ 0_{V} \}$ and $V$. Other subspaces are non-trivial. > [!Example] > For any $d\in \mathbb{N}$, $W=\mathbb{R}[x]_{\leq d}$ is a non-trivial subspace of $V=\mathbb{R}[x]$. # Properties > [!NOTE] Proposition (Intersection of Subspaces is a Subspace) > Let $V$ be a vector space and $W_{1},W_{2}\subset V$ be subspaces. Then $W_{1} \cap W_{2}$ is a subspace of $V$. >*Proof*. Since $W_{1},W_{2}$ are subspaces, $0_{V} \in W_{1} \cap W_{2}$. >Let $w_{1},w_{2}\in W_{1} \cap W_{2}$. Then $w_{1},w_{2} \in W_{1}$ so $\lambda_{1}w_{1}+\lambda_{2}w_{2}\in W_{1}$ for all $\lambda_{1},\lambda_{2}\in\mathbb{R}$. >Similarly $\lambda_{1}w_{1}+\lambda_{2}w_{2}\in W_{2}$. Hence $\lambda_{1}w_{1}+\lambda_{2}w_{2}\in W_{1} \cap W_{2}$. $\square$ > [!NOTE] Proposition (Criteria for Union of Subspace is a Subspace) > $W_{1} \cup W_{2}$ is a subspace of $V$ iff $W_{1} \subset W_{2}$ or $W_{2} \subset W_{1}$. >*Proof.* WLOG suppose $W_{1} \subset W_{2}$. Then $W_{1} \cup W_{2}=W_{2}$ is a subspace of $V$. >Conversely, suppose $W_{1} \not \subset W_{2}$ and $W_{2} \not \subset W_{1}$. Then $\exists w_{1} \in W_{1} \setminus W_{2}$ and $\exists w_{2} \in W_{2} \setminus W_{1}$. >Suppose $w_{1} + w_{2}\in W_{1}$ then $w_{1}-w_{1}+w_{2}\in W_{1} \iff w_{2}\in W_{1}$ which contradicts the fact that $w_{2} \not \in W_{1}$. Similarly, $w_{1}+w_{2} \not \in W_{2}$ hence $W_{1} \cup W_{2}$ is not a subspace. > [!NOTE] Definition (Sum of Subspaces) > If $W_{1},W_{2}\subset V$ are subspaces of a vector space $V$, then we define their *sum* to be exactly the same as their combined span $W_{1}+W_{2}= \langle W_{1} \cup W_{2}\rangle$ > [!NOTE] Lemma (Dimension of subspace of FDVS) > Any subspace of a [[Finite Dimensional Vector Space|FDVS]] $V$ has *dimension* less than or equal to $\dim V$. ^dc4bab >See [[Basis of Vector Space#^61f832|proof]]. > [!NOTE] Theorem (Dimension Formula) > Let $U_{1}, U_{2}$ be two [[Finite Dimensional Vector Space|finite dimensional]] subspaces of $V.$ Then $\dim U_{1} +\dim U_{2} = \dim (U_{1}+U_{2}) + \dim(U_{1} \cap U_{2}).$ > See [[Dimension of Sum of Finite Dimensional Vector Subspaces (Dimension Formula)|Proof]]. > [!NOTE] Definition (Complement of a subspace) > Let $V$ be a vector space and $U\subset V$ a subspace. Then a subspace $U'\subset V$ is called a **[[Complement to Vector Subspace|complement]]** to $U$ iff $V=U+U'$ and $U \cap U'=\{ 0_{V} \}.$ > [!NOTE] Definition (Orthogonal complement) > Let $V=\mathbb{R}^{n}$ equipped with [[Dot Product in Real n-Space|dot product]], and let $U\subset V$ be a [[Vector Subspace|subspace]]. We define the [[Orthogonal Complement of Subspace of Real n-Space|orthogonal complement]] $U^{\perp}$ of $U$ in $V$ to be the set of elements of $V$ that are orthogonal to every element of $U:$ $U^{T} = \{ v\in V \mid u \cdot v = 0 \quad \text{for all } u\in U \}$