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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)
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