Question

are the following statements true or false? 1. if the set of vectors u spans a subspace s, then vectors can be removed from u to create a basis for s 2. if the set of vectors u spans a subspace s, then vectors can be added to u to create a basis for s 3. if s

Answer 1

The first statement is false as removing vectors from u will not create a basis for s. The second statement is also false as additional vectors are not needed to create a basis for s. The third statement is true as the dimension of a subspace remains the same regardless of the set of vectors chosen.

Step-by-step explanation:

1. False. If the set of vectors u spans a subspace s, it already contains a basis for s. It is not possible to remove vectors from u and still have a basis for s. A basis is a set of linearly independent vectors that span the subspace.

2. False. If the set of vectors u spans a subspace s, it is not necessary to add more vectors to create a basis for s. A basis for s can be formed using the existing vectors in u.

3. True. In a subspace, the number of vectors in a basis is always the same, regardless of the particular set of vectors chosen. This number is known as the dimension of the subspace.