Label the following statements as being true or false. (a) If V is a vector space and W is a s

Jason Yuhas 2022-01-07 Answered
Label the following statements as being true or false.
(a) If V is a vector space and W is a subset of V that is a vector space, then W is a subspace of V.
(b) The empty set is a subspace of every vector space.
(c) If V is a vector space other than the zero vector space {0}, then V contains a subspace W such that W is not equal to V.
(d) The intersection of any two subsets of V is a subspace of V.
(e) An \(\displaystyle{n}\times{n}\) diagonal matrix can never have more than n nonzero entries.
(f) The trace of a square matrix is the product of its entries on the diagonal.

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Karen Robbins
Answered 2022-01-08 Author has 4657 answers
(a) If V is a vector space and W is a subset of V that is a vector space, then W is a subspace of V.
The given statement does not specify the fields over which V and W are vector spaces.
If V and W are vector spaces over the same field, then they will be closed under the operations addition and multiplication and hence the statement will be true.
(b) The empty set is a subspace of every vector space.
Note that, every vector space contains the zero vector.
Since empty set does not contain any element, empty set is not a vector space.
Hence, empty set is not a subspace.
Therefore, the statement is false.
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ambarakaq8
Answered 2022-01-09 Author has 1384 answers
(c) If V is a vector space other than the zero vector space {0}. then V contains a subspace W such that W is not equal to V.
The zero vector of a subspace is always the same as the zero vector of the whole space.
That is, 0=V-V for any vector v.
Consider any vector space V and its subspace W.
If w \(\displaystyle\in\) W, the zero vector of W is w-w which is also the zero vector of V.
Thus, the statement 15 true.
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