Subspaces Quiz

What is a vector subspace?

• A type of musical instrument

• A subset of a vector space that is closed under addition and scalar multiplication.

• A type of cooking utensil

• A type of geometric shape

If a set of vectors is linearly independent, what can you say about the scalars c1, c2, ..., cn in the linear combination c1v1 + c2v2 + ... + cnvn = 0?

• The scalars c1, c2, ..., cn must all be equal to 1.

• The scalars c1, c2, ..., cn must all be equal to -1.

• The scalars c1, c2, ..., cn must all be equal to 2.

• The scalars c1, c2, ..., cn must all be equal to 0.

If a set of vectors spans a vector space V, what can you say about any vector in V?

• Any vector in V is independent of the vectors in the set.

• Any vector in V can be expressed as a linear combination of the vectors in the set.

• Any vector in V is orthogonal to the vectors in the set.

• Any vector in V cannot be expressed as a linear combination of the vectors in the set.

True or False: The zero vector is always in a vector subspace.

• True

• False

• Sometimes

• Depends on the dimension

What is the dimension of the subspace spanned by the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1)?

• 3

• 2

• 4

• 1

If a set of vectors spans a vector space V, is it possible for the set to be linearly dependent?

• Maybe

• No

• Depends on the dimension of V

• Yes

What is the dimension of the subspace spanned by the vectors (1, 2, 3) and (2, 4, 6)?

• 1

• 4

• 3

• 2

True or False: The intersection of two subspaces is always a subspace.

• Depends on the dimensions

• Sometimes

• True

• False

If a set of vectors is linearly independent and spans a vector space V, what can you say about the set?

• It is not a subset of the vector space V

• It is a subset of the vector space V

• It is a linearly dependent set

• It forms a basis for the vector space V.

What is the dimension of the subspace spanned by the vectors (1, 1, 1) and (1, -1, 0)?

• 2

• 3

• 4

• 1