Vector subspace

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. 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, y ∈ W implies x + y ∈ W
3. 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]\displaystyle{ W_1 }[/math] and [math]\displaystyle{ W_2 }[/math] are subspaces of a vector space [math]\displaystyle{ V }[/math], then the sum and the direct sum of [math]\displaystyle{ W_1 }[/math] and [math]\displaystyle{ W_2 }[/math], denoted respectively by [math]\displaystyle{ W_1 + W_2 }[/math] and [math]\displaystyle{ W_1 \oplus W_2 }[/math], are subspaces as well.

Vector Subspace Media

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  The projective line F5P1 over a finite field F5.

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  The projective line F5P1 over a finite field F5.

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  The projective line F5P1 over a finite field F5.

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  The projective line F5P1 over a finite field F5.

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  The line is the set of W. W lies within V (the plane). W is a subspace. Made with GNU Plot.

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  The vectors u and v are a basis for this two-dimensional subspace of R3.

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  In R3, the intersection of two distinct two-dimensional subspaces is one-dimensional

Related pages

• Parallel projection