From 32f4af5f369fa9f0b2988ecad7797f4bec3661c3 Mon Sep 17 00:00:00 2001 From: Holden Rohrer Date: Tue, 21 Sep 2021 17:12:46 -0400 Subject: notes and homework --- li/04_vector_space | 30 ++++++++++++++++++++++++++++++ 1 file changed, 30 insertions(+) create mode 100644 li/04_vector_space (limited to 'li/04_vector_space') diff --git a/li/04_vector_space b/li/04_vector_space new file mode 100644 index 0000000..720f44b --- /dev/null +++ b/li/04_vector_space @@ -0,0 +1,30 @@ +Vector Space is a set V and binary addition operator and a +scalar-by-vector multiplication operator where V is closed under the +binary and multiplication operators. (V, +, *) denotes the space. +V = R^n is one example of a vector space. + +A subspace is a subset of V s.t. (S, +, *) is a closed vector space. + +With A an mxn matrix, + +S = { x in R^n | Ax = 0 } is a subspace in R^n + +It is sufficient proof to check: +(i) \forall x,y \in S, x+y \in S + If Ax = 0, Ay = 0, A(x+y) = Ax + Ay = 0 + 0 = 0. +(ii) \forall x\in S, \alpha\in R, \alpha x \in S. + A(\alpha*x)= \alpha(Ax) = \alpha*0 = 0. +Therefore, S is a subspace of R^n. + +This is called the null space of A, null(A). + +AX = 0 is also a vector space over X\in R^M by R^M. + +If A is invertible, AX = 0 \to A^{-1} AX = A^{-1} 0 \to X = 0. + +S = span{a1, a2, ..., a_n} is a subspace in R^m + Proof of subspace omitted. + +S = column space of A = C(A) = R(A) = span of A. +R(A) stands for range. +Ax = b has a solution, iff b \in R(A) -- cgit