A **vector space** is a set $V$ of **vectors** that can be [[abelian group|added]] to each other and scaled (by elements of some [[field]] $\mathbb{F}$). the scalar multiplication satisfies $a(bv) = (ab)v$ and $(a + b) v = av + bv$ and $a(x + y) = ax + y$ and has $1v = v$. Then $0v = v$ and $\alpha 0 = 0$ and $(-1)u = -u$ If there's also vector-vector multiplication, we get an [[algebra]] a [[homomorphism]] of vector spaces is a [[linear]] map. # sources [[2015AxlerLinearAlgebraDone|Axler]] 1.19-1.31 ![[bilinear form visualization.excalidraw|200]] [[2015AxlerLinearAlgebraDone|Axler]] [Interactive Linear Algebra](https://textbooks.math.gatech.edu/ila/overview.html) [x.com](https://x.com/michael_nielsen/status/1795920578769039463) [[3B1B Wordle video]]