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

Answer:

a) True

b) False

c) False

d) False

e) True

Step-by-step 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 $V\neq span(A)$ 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.