Question

When we prove that a subspace is closed under addition and multiplication by a scalar are we proving that such subspace is a vector space?

Answer 1

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Answer 2

Yes. :)

Answer 3

No. The characteristics of a vector space are that it is closed under addition and multiplication by a scalar. A subspace is a vector space that's inside another, larger, vector space. An example would be the R³ and a 2-dimensional plane that's spanned by 2 different vectors in R³. All the elements inside the plane ( that are represented as linear combinations of the vectors that span it ) are closed under addition and multiplication by a scalar. All the elements inside R³ are also closed under addition and mult. by a scalar. All the elements that are contained in the plane are also inside R³. Then, you have proved that the plane is a subspace of R³.

Answer 4

When a subset is closed under addition and multiplication by scalar, then we can say that it's a subspace of a vector space.

Answer 5

Most textbooks on linear algebra state a theorem which says that if a subset of a vector space is closed under the vector addition and scalar multiplication of the big vector space, then the subset is a vector space and hence is a vector subspace.

Strang generally does not state and prove theorems, but the basic ideas of this theorem are discussed on page 122 of the fourth edition. See the second paragraph below the shaded box.