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If W is a set of one or more vectors from a vector space V , then W is a subspace of V if and only if the following conditions hold. (a) If u and v are vectors in W, then u + v is in W. (b) If k is any scalar and u is any vector in W, then ku is in W.
0:01 4:16 Determine if W is a Subspace of a Vector Space V - YouTube YouTube Start of suggested clip End of suggested clip The second condition is that W is closed under vector addition. So if you take two vectors x and yMoreThe second condition is that W is closed under vector addition. So if you take two vectors x and y in W. Then whenever this happens the sum X plus y must also be in W.
A nonempty subset W of a vector space V is a subspace of V if W is closed under addition and scalar multiplication. If a subset S of a vector space V does not contain the zero vector 0 , then S cannot be a subspace of V .
Given subspaces U and W of a vector space V, then their intersection U W := {v V : v is an element of both U and W} is also a subspace of V. Proof: Let v and w be elements of U W. Then v and w belong to both U and W.
The multilinear maps form a vector space Mult(V1,,Vk;W). Multilinear maps in k variables are also called k-linear. The 1-linear maps are the linear maps, the 2-linear maps are the bilinear maps.
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For A(W) to be a subspace of V it must be closed under addition and scalar multiplication over V. V means the dual space of V which means the set of all linear maps ϕ:Vk.

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