Question

Consider a linear space V. For which linear transformations Tfrom Vto is

$$\begin{gathered} {\mathit{R}}^{\mathbf{n}}\mathbf{}\mathit{i}\mathit{s} \\ <v,w>\mathbf{=}\mathit{T}\left(v\right)\mathbf{.}\mathit{T}\left(w\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\uparrow }\mathbf{}\mathbf{} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{D}\mathit{o}\mathit{t}\mathbf{}\mathit{p}\mathit{r}\mathit{o}\mathit{d}\mathit{u}\mathit{c}\mathit{t} \end{gathered}$$

an inner product in V?

Short answer

T is one-one.

Step-by-step solution

Check the properties.

Here is V is a vector space and we want to find liner transformation $T:V\to R^n$such that

is an inner product.

Let $u,v,w\in V$and $a\in R$.

Then, consider the following properties.

1)

2)

3)

4) Let$v\ne 0$

The above term is not a non-zero.

Hence, the transformation T is one-one.