Jason Yuhas

2022-01-07

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$n\times n$ 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.

(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

(f) The trace of a square matrix is the product of its entries on the diagonal.

Karen Robbins

Beginner2022-01-08Added 49 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.

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.

ambarakaq8

Beginner2022-01-09Added 31 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$\in$ W, the zero vector of W is w-w which is also the zero vector of V.

Thus, the statement 15 true.

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

Thus, the statement 15 true.

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