Explanation → Linear transformationsEvidenceCandidate 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.
Content location → MatricesEvidenceCandidate from reviewed en material v1: A linear map with input and output spaces of the same dimension has a square matrix representation once bases are chosen. The earlier examples use 2×2 and 3×3 matrices.
Content location → MatricesEvidenceCandidate from reviewed zh material v1: 3×2 矩阵表示从二维输入空间到三维输出空间的线性映射。其非方形状记录了不同数量的输入和输出坐标。
Content location → Linear transformationsEvidenceCandidate 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。
Explanation → Nonsquare matrices as transformations between dimensions | 3Blue1BrownEvidenceCandidate 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.
Explanation → Linear transformationsEvidenceCandidate 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。
Explanation → MatricesEvidenceCandidate from reviewed en material v1: A linear map with input and output spaces of the same dimension has a square matrix representation once bases are chosen. The earlier examples use 2×2 and 3×3 matrices.
Explanation → MatricesEvidenceCandidate from reviewed zh material v1: 3×2 矩阵表示从二维输入空间到三维输出空间的线性映射。其非方形状记录了不同数量的输入和输出坐标。
Content location → Nonsquare matrices as transformations between dimensions | 3Blue1BrownEvidenceCandidate from reviewed en material v1: A linear map with input and output spaces of the same dimension has a square matrix representation once bases are chosen. The earlier examples use 2×2 and 3×3 matrices.
Explanation → MatricesEvidenceCandidate from reviewed en material v1: A linear map with input and output spaces of the same dimension has a square matrix representation once bases are chosen. The earlier examples use 2×2 and 3×3 matrices.
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