Quartz sync: Mar 1, 2025, 2:26 PM

content/Definitions/Vector Spaces/Complex Vector Space.md

@@ -0,0 +1,2 @@
+# Definition
+A **complex vector space** is a set $V$ with addition $u + v$ of vectors $u$, $v$, and scalar multiplication $a \times v, \; a \in \mathbb{C}, \; v \in V$, satisfying the [[Properties of a Vector Space]]

content/Definitions/Vector Spaces/Linear Basis.md

@@ -0,0 +1,2 @@
+# Definition
+A **linear basis** of $V$ is a subset $\{ v_{i} \} \subset V$ such that every vector can be written uniquely as $\Sigma_{i} c_{i} v_{i}$ for finitely many non-zero $c_{i} \in \mathbb{C}$.

content/Definitions/Vector Spaces/Normed Vector Space.md

@@ -0,0 +1,2 @@
+# Definition
+Have a metric given by the [[Norm]] on a vector space $V$ as $d(u, v) \equiv \|u - v\| \in [ \, 0, \infty \rangle$.

content/Definitions/Vector Spaces/Properties of a Vector Space.md

@@ -0,0 +1,10 @@
+- $u + v = v + u$,
+- $(u+v) + w = u + (v + w)$,
+- $u + 0 = u$,
+- $u + (-u) = 0$,
+- $a (u + v) = a \times u + a \times v$,
+- $(a + b) \times v = av + bv$,
+- $a(bv) = (ab) \times v$,
+- $1 \times v = v$.
+
+These can be used for Complex, Real, or Rational vector spaces.