channelflow.org

**This is an old revision of the document!** ----

====== IAM 961 HW1 ====== **1** Prove that any linear map $\mathcal{L} : \mathbb{C}^n \rightarrow \mathbb{C}^m$ can written as an $m \times n$ matrix. (Hint: let $y = \mathcal{L}(x)$. Express $x$ as a linear combination of the canonical basis vectors $\{e_j\}$. Substitute that into $y = \mathcal{L}(x)$, then use linearity to rewrite the right-hand-side of this equation as a linear combination of vectors $\mathcal{L}(e_j)$. Now take the inner product of both sides of this equation with $e_i$. That should give you $y_i = \sum_j=1^n L_{ij} x_j$ for some matrix $L$.)