Let U and W be subspace of a vector space V show that (1) U+W is subspace of v (2) U and W are contained in U+W (3) W+W=W

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To prove the given statements about the subspaces U and W of a vector space V, let's go through each one:
1) To show that U + W is a subspace of V, we need to verify three conditions: closure under addition, closure under scalar multiplication, and containing the zero vector.
Closure under addition: Let

𝐮_{1}

and

𝐮_{2}

be vectors in U, and let

𝐰_{1}

and

𝐰_{2}

be vectors in W. Since U and W are subspaces,

𝐮_{1} + 𝐮_{2}

is in U and

𝐰_{1} + 𝐰_{2}

is in W. Therefore,

(𝐮_{1} + 𝐰_{1}) + (𝐮_{2} + 𝐰_{2})

is in U + W.
Closure under scalar multiplication: Let

𝐯

be a vector in U + W, and let

c

be a scalar. Since

𝐯

is a combination of vectors from U and W,

c 𝐯

is a combination of vectors from U and W as well. Therefore,

c 𝐯

is in U + W.
Containing the zero vector: Since U and W are subspaces, they contain the zero vector. Therefore, the zero vector is in U + W.
Hence, U + W satisfies all the conditions to be a subspace of V.
2) To show that U and W are contained in U + W, we need to prove that every vector in U is also in U + W and every vector in W is also in U + W.
Let

𝐮

be a vector in U. Since U is a subspace,

𝐮

can be written as

𝐮 + 0

, where

𝐮

is in U and

0

is in W. Therefore,

𝐮

is in U + W.
Similarly, let

𝐰

be a vector in W. Since W is a subspace,

𝐰

can be written as

0 + 𝐰

, where

0

is in U and

𝐰

is in W. Thus,

𝐰

is in U + W.
Hence, U and W are contained in U + W.
3) To prove W + W = W, we need to show that every vector in W + W is also in W and every vector in W is in W + W.
Let

𝐰_{1}

and

𝐰_{2}

be vectors in W. Since W is a subspace,

𝐰_{1} + 𝐰_{2}

is in W. Therefore, every vector in W + W is in W.
Now, let

𝐰

be a vector in W. Since W is a subspace,

𝐰 = 𝐰 + 0

, where

0

is in W. Hence, every vector in W is in W + W.
Therefore, W + W = W.
By proving these statements, we have shown that U + W is a subspace of V, U and W are contained in U + W, and W + W = W.

Let U and W be subspace of a

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