Question

Let V be a vector space of dimension 4. Determine if each statement is true or false. (a) Any set of 5 vectors in V must be linearly dependent. (b) Any set of 5 vectors in V must span V (c) Any set of 4 nonzero vectors in V must be a basis for V. (d) Any set of 3 vectors in V must be linearly independent (e) No set of 3 vectors in V can span V.

Answer 1

a) True

b) False

c) False

d) False

e) True

Explanation:

a) Each basis of V has four vectors. Then any set of 5 vectors must be linear dependent (LD).

b) Suppose that
`\{v_1,v_2,v_3,v_4\}` is a basis of V. Considere the set
`A=\{v_1,\lambda_1v_1,\lambda_2v_1,v_2,v_3\}` where
`\lambda_1, \lambda_2` are scalars. The set has 5 vectors but
`V\\eq span(A)` because
`v_4` is not belong to A and
`v_4` is linear independent of
`v_1`

c) Suppose that
`\{v_1,v_2,v_3,v_4\}` is a basis of V. Considere the set
`A=\{v_1,\lambda_1v_1,\lambda_2v_1,\lambda_3v_1\}` where
`\lambda_1, \lambda_2,\lambda_3` are scalars. A has four nonzero vectors but isn't a basis because is a LD set.

d) Suppose that
`\{v_1,v_2,v_3,v_4\}` is a basis of V. Considere the set
`A=\{v_1,\lambda_1v_1,\lambda_2v_1\}` where
`\lambda_1, \lambda_2` are scalars. A has 3 nonzero vectors but isn't a basis because is a LD set.

e) Since any basis of V must have 4 elements, then a set of three vectors cannot generate V.