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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:
- additive identity – the element 0 is an element of W: 0 ∈ W
- closed under addition – if x and y are elements of W, then x + y is also in W: x, y ∈ W implies x + y ∈ W
- closed under scalar multiplication – if c is an element of a field K and x is in W, then cx is in W: c ∈ K and x ∈ W implies cx ∈ W.
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], are subspaces as well.
- Axler, Sheldon (2015). Linear Algebra Done Right (Third ed.). Springer International Publishing. p. 18. . .
- "Subspace | Brilliant Math & Science Wiki" (in en-us). https://brilliant.org/wiki/subspace/.
- "Comprehensive List of Algebra Symbols" (in en-US). 2020-03-25. https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/.
- "4.4: Sums and direct sum" (in en). 2013-11-07. https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/04%3A_Vector_spaces/4.04%3A_Sums_and_direct_sum.