# Linear maps and vectors.

• Sep 7th 2009, 02:00 PM
sean.1986
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.
• Sep 7th 2009, 03:31 PM
NonCommAlg
Quote:

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 $f: \mathbb{R}^n \longrightarrow \mathbb{R}$ by $f(v)=u^T v.$

Quote:

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