Question

If V is a nonzero element of V, prove that $\left\{v\right\}$is linearly independent over F.

Short answer

Answer:

$\left\{v\right\}$is linearly independent over F.

Step-by-step solution

Definition of linearly independent vector.

In a vector space , a set of the vector is said to be linearly dependent if one of the vectors in the set can be written as a linear combination of other, and if the no vectors can be written as a linear combination of others then the vector is said to be linearly independent.

Step 2:Showing that {v} is linearly independent.

Let v is a nonzero element of a vector space Vover F.

If $b·v=0$

Multiply both side by $b^{-1}$.

$$\begin{gathered} b^{-1}bv=b^{-1}{0}_v \\ 1.v=0_v \end{gathered}$$

Which is a contradiction as $1·v=v$ and vis a nonzero element of a vector space V.

Therefore, role="math" localid="1656918461827" $b·v\ne 0$ for any nonzero b.

Thus $\left\{v\right\}$ is linearly independent over F.