Linear Algebra — 29.3: How a Linear Map Becomes a Matrix

Every matrix is a linear transformation written in a chosen basis. This video unpacks the column rule that builds the matrix of any linear map — and shows why the same transformation can wear completely different matrix faces depending on the basis you pick.

Starting with two very different matrices that both describe projection onto the 45° line, we derive the universal recipe: apply the transformation to each input basis vector, write the result in coordinates of the output basis, and stack the results as columns. The same rule then builds the matrix of the derivative on polynomials, revealing calculus as linear algebra in disguise.

Key concepts covered:
• Linear transformations vs. their matrix representations
• The column rule: column j of A equals T(v_j) written in the output basis
• Why the rule works — linearity extends agreement on a basis to all vectors
• Projection onto a line in two different bases (eigenvector basis vs. standard basis)
• Diagonal matrices and eigenvalues as the "natural" coordinates of a transformation
• The derivative as a 2×3 matrix acting on polynomials of degree at most two
• Why matrix multiplication encodes function composition
• Three takeaways: matrix = transformation + basis, eigenvectors diagonalize, calculus is linear

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SOURCE MATERIALS
The source materials for this video are from https://www.youtube.com/watch?v=Ts3o2I8_Mxc

Видео Linear Algebra — 29.3: How a Linear Map Becomes a Matrix канала Ludium