Introduction to Linear Algebra

Linear combination

Let $\mathbf{v},\mathbf{w}\in {\mathbb{R}}^m$ $\lambda \mathbf{v}+\mu \mathbf{w}\in {\mathbb{R}}^m$

Linear Algebra ETH, p.16

Every vector in ${\mathbb{R}}^2$ is a linear combination of the two vectors $\mathbf{v}=\left(\begin{array}{c}2\\ 3\end{array}\right),\mathbf{u}=\left(\begin{array}{c}3\\ -1\end{array}\right)$

Proof trivial, $\lambda \left(\begin{array}{c}2\\ 3\end{array}\right)+\mu \left(\begin{array}{c}3\\ -1\end{array}\right)=\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)$ Both $\lambda$ and $\mu$ can be solved in terms of $u_1$ and $u_2$

special linear combination

• affine: ${\lambda }_1+{\lambda }_2+\cdots +{\lambda }_n=1$
  • $\lambda +\mu =1$
  • $\lambda \mathbf{v}+\mu \mathbf{w}$
  • $\lambda \mathbf{v}+\mathbf{w}-\lambda \mathbf{w}$
  • $\mathbf{w}+\lambda (\mathbf{v}-\mathbf{w})$
• conic: $\lambda \ge 0$ for $j=1,2,\dots ,n$
• convex: if it is both a affine and a conic combination