Question

let $\left\{V_1,.......V_n\right\}$be as basic of $\mathit{V}$over $\mathit{F}$let ${\mathit{c}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathit{c}}_{\mathbf{n}}$be nonzero element of$\mathit{F}$, then prove that $\mathbf{\{}{\mathbf{c}}_{\mathbf{1}}{\mathbf{v}}_{\mathbf{1}}\mathbf{,}{\mathbf{c}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{c}}_{\mathbf{n}}{\mathbf{v}}_{\mathbf{n}}\mathbf{\}}$is also a basic of $\mathit{V}$over$\mathit{F}$.

Short answer

Answer:

$\mathbf{\{}{\mathbf{c}}_{\mathbf{1}}{\mathbf{v}}_{\mathbf{1}}\mathbf{,}{\mathbf{c}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{c}}_{\mathbf{n}}{\mathbf{v}}_{\mathbf{n}}\mathbf{\}}$is a basic of V over F.

Step-by-step solution

Definition of linearly dependent.

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

Step 2:Showing {c1v1,c2v2,....cnvn} is a basis.

Letbe a basic of V over F and be nonzero element of F.

It is enough to show that the setis linearly independent and spans V over F.

Sinceis a basic of V over F.

Thereforeis linearly independent and spans V over F.

This implies that there exists scalarssuch thatand all

Now let

Asis linearly independent.

Therefore it givesas

Therefore all

Since all the scalers are zero. Thus the setis linearly independent.

For span

Since the setis a span V over F.

Therefore for, it can be written as a linear combination of element of the set.

This implies that

Then

Let. Then this implies,

This states thatcan be written as a linearly combination of element of the set.

Therefore is linearly independent and spans V overF.

Thus $\mathbf{\{}{\mathbf{c}}_{\mathbf{1}}{\mathbf{v}}_{\mathbf{1}}\mathbf{,}{\mathbf{c}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{c}}_{\mathbf{n}}{\mathbf{v}}_{\mathbf{n}}\mathbf{\}}$is a basic for V over F.