If U and V are subspaces of a vector space V then,
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Both U,W are subspaces of V, which tells us that 0 ∈ U and 0 ∈ W, which means 0 ∈ U ∩ W. If u, w ∈ U ∩ W, then by definition u, w ∈ U and u, w ∈ W. Since U,W are subspaces, they are closed under addition - meaning u + w ∈ U and u + w ∈ W, implying u + w ∈ U ∩ W. ... Thus the intersection U ∩ W is a subspace of V.
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Both U,W are subspaces of V, which tells us that 0 ∈ U and 0 ∈ W, which means 0 ∈ U ∩ W. If u, w ∈ U ∩ W, then by definition u, w ∈ U and u, w ∈ W. Since U,W are subspaces, they are closed under addition - meaning u + w ∈ U and u + w ∈ W, implying u + w ∈ U ∩ W. ... Thus the intersection U ∩ W is a subspace of V.
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