> [!NOTE] Definition (Basis) > A subset $S$ of a [[Vector spaces|finite dimensional vector space]] $V$ is basis of $V$ iff it is a [[Span of Subset of Vector Space#^5c027b|spanning set]] of $V$ that is [[Linear Independence|linearly independent]]. ^cbe5c8 > [!NOTE] Definition (Orthonormal Basis) > A Basis that is [[Orthonormal Subset of Euclidean Space|orthonormal]]. >Note [[Gram-Schmidt orthogonalisation in real n-space]]. # Properties > [!NOTE] Proposition (Unique linear combination of basis for each vector) > Let $\underline{f}_{1},\dots,\underline{f}_{n}$ be a basis of $V$. Then for $\underline{w}\in V$, there are unique scalars $\mu_{1},\dots,\mu_{n}\in\mathbb{R}$ such that $\underline{w}=\mu_{1}\underline{f}_{1}+\dots+\mu_{n}\underline{f}_{n}.$ >*Proof*. Since $\underline{f}_{1},\dots,\underline{f}_{n}$ span $\mathbb{R}^n$, > [!NOTE] Corollary (Every FDVS has a basis) > Let $V$ be a FDVS and $S\subset V$ a linearly independent subset. Then there exists $T\subset V$ so that $S\cup T$ is basis. >*Proof*. Since $V$ is finite dimensional, there is a subset $M\subset V$ that spans $V$. Apply sifting lemma with $L$ and $S=L\cup M$. The following lemma allows us to prove that any two bases (which will be finite by their [[Linear Independence#^fbfe78|linear independence]]) have the same cardinality. > [!Example] > Let $B=\{ \underline{i},\underline{j},\underline{k} \}$ be the standards basis of $\mathbb{R}^{3}$ and consider $\underline{w}=(2,-3,0)^{T}=2\underline{i}-3\underline{j}$ > The exchange lemma says that $\{ \underline{w},\underline{j},\underline{k} \},\{ \underline{i},\underline{w},\underline{k} \}$ are both bases of $\mathbb{R}^{3}$. > [!NOTE] Theorem (Size of any two bases is the same) > If $V$ is a finite-dimensional vector space (over $\mathbb{R}$ for example), then any two bases are finite and have the same size. ^b1e1c3 > [!NOTE] Definition (Dimension) > Let $V$ be a FDVS. The dimension of $V$ is given by $\dim V= \text{ \# any basis of V.}$ ^db42e0 > [!NOTE] Corollary > Let $W \subset V$ be a subspace of a FDVS. If $S\subset W$ is linearly independent with $|S|\geq\dim V$, then $W=V$. # Applications - [[Matrix representations of linear map]]