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Hub AI
Linear subspace AI simulator
(@Linear subspace_simulator)
Hub AI
Linear subspace AI simulator
(@Linear subspace_simulator)
Linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.
If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V. Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w1, w2 are elements of W and α, β are elements of K, it follows that αw1 + βw2 is in W.
The singleton set consisting of the zero vector alone and the entire vector space itself are linear subspaces that are called the trivial subspaces of the vector space.
In the vector space V = R3 (the real coordinate space over the field R of real numbers), take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V.
Proof:
Let the field be R again, but now let the vector space V be the Cartesian plane R2. Take W to be the set of points (x, y) of R2 such that x = y. Then W is a subspace of R2.
Proof:
In general, any subset of the real coordinate space Rn that is defined by a homogeneous system of linear equations will yield a subspace. (The equation in example I was z = 0, and the equation in example II was x = y.)
Linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.
If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V. Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w1, w2 are elements of W and α, β are elements of K, it follows that αw1 + βw2 is in W.
The singleton set consisting of the zero vector alone and the entire vector space itself are linear subspaces that are called the trivial subspaces of the vector space.
In the vector space V = R3 (the real coordinate space over the field R of real numbers), take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V.
Proof:
Let the field be R again, but now let the vector space V be the Cartesian plane R2. Take W to be the set of points (x, y) of R2 such that x = y. Then W is a subspace of R2.
Proof:
In general, any subset of the real coordinate space Rn that is defined by a homogeneous system of linear equations will yield a subspace. (The equation in example I was z = 0, and the equation in example II was x = y.)