Question

Question: Prove that $\left\{v_1,.........v_n\right\}$is a basis of over if and only if every element of can be written in a unique way as a linear combination of$\mathbf{\{}{\mathbf{v}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{v}}_{\mathbf{n}}\mathbf{\}}$ (“ Unique” means that if$\mathit{\omega }\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{v}}_{\mathbf{1}}\mathbf{+}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{+}{\mathit{c}}_{\mathbf{n}}{\mathit{v}}_{\mathbf{n}}$ androle="math" localid="1658825939686" $\mathit{\omega }\mathbf{=}{\mathit{d}}_{\mathbf{1}}{\mathit{v}}_{\mathbf{1}}\mathbf{+}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{+}{\mathit{d}}_{\mathbf{n}}{\mathit{v}}_{\mathbf{n}}$ then for${\mathit{c}}_{\mathbf{i}}\mathbf{=}{\mathit{d}}_{\mathbf{i}}$ every i)

Short answer

Answer

Thus, the representation ofv is unique only if part.

Step-by-step solution

Explanation

Here we prove if part .

Let$\left\{v_j,.........v_n\right\}$ is a basis for V, then we have to show that every element of V can be written as a unique linear combination$\left\{v_j,.........v_n\right\}$.

Concept

Let an element can be written as linear combination of$\left\{v_j,.........v_n\right\}$ in two ways .

$v=a_j{v}_j+.....+a_n{v}_n$____________(1)

_role="math" localid="1658826261934" $v=b_j{v}_j+.....+b_n{v}_n$__________(2)

Solution

From (i) and (2)

$$\begin{gathered} \Rightarrow a_j{v}_j+.....+a_n{v}_n=b_j{v}_j+.....+b_n{v}_n \\ \Rightarrow a_j{v}_j+.....+a_n{v}_n-\left(b_j{v}_j+.....+b_n{v}_n\right)=0 \\ \Rightarrow \left(a_j-b_j\right)v_j+---+\left(a_n-b_n\right)v_n=0 \end{gathered}$$

Since is basis.

$$\begin{gathered} \therefore a_j-b_j=0,----,a_n-b_n=0 \\ \Rightarrow a_j=b_j---,a_n=b_n \end{gathered}$$

Hence representation of v is unique only if part.