Proving W is a Subspace of V: Let u & v be Vectors in V

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The discussion focuses on proving that the set W, defined as all linear combinations of two fixed vectors u and v in a vector space V, is a subspace of V. The proof involves demonstrating that any linear combination of vectors from W remains within W. Specifically, if two vectors from W are taken, represented as au + bv and cu + dv, their linear combination x(au + bv) + y(cu + dv) must also be expressible in the form of a linear combination of u and v, thereby satisfying the subspace criteria.

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Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations au+bv of u and v is a subspace of V.

I cannot prove the above proof properly. Can anyone help.

-Thanks
 
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ECE said:
Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations au+bv of u and v is a subspace of V.

I cannot prove the above proof properly. Can anyone help.

-Thanks

Take two vectors from W, and show that their linear combination is also in W.
 
"Take two vectors from W" means taking two linear combinations, perhaps with different "a" and "b', say au+ bv and cu+ dv. "Their linear combination" would be something like x(au+ bv)+ y(cu+ dv) for numbers, x and y. "Show it is also in W" means "show it satisfies the definition of W". Here that means show that it also can be written au+ bv for some choices of a and b.
 
Thanks i understand it now
 

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