# Thread: Linear maps and vectors.

1. ## Linear maps and vectors.

A linear map L: R^n -> R is linear for all v, w that exist in R^n and all a, b that exist in R if...

L(av + bw) = aL(v) + bL(w)

Show that for any vector u that exists in R^n, the transposed vector u^T represents a linear map of type R^n -> R

Also, prove conversely that a linear map L:R^n -> R can be represented as the transpose u^T of a vector u that exists in R^n.

Sorry for the lack of latex usage. I don't know all the symbols and I'm hoping for an answer as soon as possible.

Sean.

2. Originally Posted by sean.1986
A linear map L: R^n -> R is linear for all v, w that exist in R^n and all a, b that exist in R if...

L(av + bw) = aL(v) + bL(w)

Show that for any vector u that exists in R^n, the transposed vector u^T represents a linear map of type R^n -> R
define $\displaystyle f: \mathbb{R}^n \longrightarrow \mathbb{R}$ by $\displaystyle f(v)=u^T v.$

Also, prove conversely that a linear map L:R^n -> R can be represented as the transpose u^T of a vector u that exists in R^n.

Sorry for the lack of latex usage. I don't know all the symbols and I'm hoping for an answer as soon as possible.

suppose $\displaystyle f: \mathbb{R}^n \longrightarrow \mathbb{R}$ is a linear map. let $\displaystyle e_j \in \mathbb{R}^n$ be a vector with $\displaystyle 1$ in the $\displaystyle j$th row and $\displaystyle 0$ everywhere else. note that every $\displaystyle v \in \mathbb{R}^n$ can be written (uniquely) as $\displaystyle v=\sum_{j=1}^n v_je_j,$ where $\displaystyle v_j \in \mathbb{R}.$
let $\displaystyle u= \sum_{j=1}^n f(e_j)e_j.$ then since $\displaystyle f$ is $\displaystyle \mathbb{R}-$linear, we have: $\displaystyle f(v)=f \left(\sum_{j=1}^n v_je_j \right)=\sum_{j=1}^n v_jf(e_j)=\begin{bmatrix} f(e_1) & . & . & . & f(e_n) \end{bmatrix} \begin{bmatrix} v_1 \\ . \\ . \\ . \\ v_n \end{bmatrix} = u^T v.$