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Linear transformations

A linear transformation preserves vector addition and scalar multiplication; its matrix depends on the chosen bases.

Different ways to understand it

Nonsquare matrices as transformations between dimensions | 3Blue1Brown
3Blue1Brown · en · Explanation
Connection evidence
Candidate from reviewed zh material v1: 分屏示意图把三维输入映到二维平面。三个基向量像的箭头说明如何确定这种跨维映射；箭头位置不应当被视为前面数值矩阵的计算结果。
Nonsquare matrices as transformations between dimensions | 3Blue1Brown
3Blue1Brown · en · Explanation
Connection evidence
Candidate from reviewed en material v1: A nonsquare matrix represents a linear map between coordinate spaces with different dimensions. Count its columns to find the input dimension and its rows to find the output dimension. The lesson constructs a 3×2 matrix from two basis images, interprets its column space as a plane in three-dimensional space, and explains full column rank in that example. It then contrasts 2×3 and 1×2 maps and reads the row matrix [1,2] from basis vectors landing on the number line. Separate dimensional diagrams are illustrations rather than computations with one shared matrix. Dot products and duality appear as a closing preview.

LEARNING MAP

Connect vectors, linear transformations and matrix computations, with geometric and algebraic viewpoints.