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Vector subspace

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A vector subspace is a vector space that is a subset of another vector space. This means that all the properties of a vector space are satisfied. Let W be a non empty subset of a vector space V, then, W is a vector subspace if and only if the next 3 conditions are satisfied:[1][2]

  1. additive identity – the element 0 is an element of W: 0 ∈ W
  2. closed under addition – if x and y are elements of W, then x + y is also in W: x, yW implies x + yW
  3. closed under scalar multiplication – if c is an element of a field K and x is in W, then cx is in W: cK and xW implies cxW.

If [math]W_1[/math] and [math]W_2[/math] are subspaces of a vector space [math]V[/math], then the sum and the direct sum of [math]W_1[/math] and [math]W_2[/math], denoted respectively by [math]W_1 + W_2[/math] and [math]W_1 \oplus W_2[/math],[3] are subspaces as well.[4]

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