Can anyone explain why (linear algebra)

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SUMMARY

The discussion centers on the mathematical concept that the union of two subspaces V and W in R^n is not itself a subspace. Specifically, it highlights that for subspaces V = {(x,y) | y = x} and W = {(x,y) | y = -x} in R^2, the vector sum of elements from each subspace, v = (1, 1) and w = (-1, 1), does not belong to the union V ∪ W. This illustrates that the closure under addition, a fundamental property of subspaces, is violated in this case.

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Why for the subspaces V and W in R^n , the

union of V and W is not a subspace of R^n?

Thank you.
 
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Look at some examples to see why such as the union of two lines through the origin in R^2.
 
If v is in subspace V and w is in subspace W, it is not, in general, true that v+ w is in V\cupW. You can give a specific counter example by using Vid's suggestion. Let V= {(x,y)| y= x}, v= (1, 1), W= {(x,y)|y= -x}, w= (-1, 1). Is v+ w in the union of V and W?
 

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