Question

let b and d be distinct nonzero real number and c any real number, prove that $\left\{b,c+d\right\}$is a basic of $\mathit{C}$over $\mathit{R}$.

Short answer

Answer:

$\left\{b.c+d\right\}$is a basic of $\mathit{C}$over$\mathit{R}$.

Step-by-step solution

Definition of basis.

A set of element in vector space$V$ is called basic or a set of basic vectors, if the vectors are linearly independent and every vector in vector space is linear combination of this set.

Step 2:Showing that {b,c+d} is a basis of C

Let a and d be distinct nonzero real numbers and c any real number.

Set $\left\{b,c+id\right\}$will be a basic of $C$over $R$if and only if any element of $C$can be written as a linear combination of this set.

Now if $x+iy\in C,\alpha \beta \in R$

Then,

$$\begin{gathered} x+iy=\alpha \left(b\right)+\beta \left(c+id\right) \\ =\alpha \beta +\beta c+\left(\beta di\right) \end{gathered}$$

Compare the value and find scalars $\alpha ,\beta$.

$$\begin{gathered} y=\beta d \\ \beta =\frac{y}{d} \end{gathered}$$

And $\alpha \beta +\beta c=x$

Put the value of $\alpha$and find the value of $\beta$.

Hence proved.