American Mathematical Monthly · 1993 · 201 citations · 1 references
Natural SciencesMathematicsAlgebraLinear AlgebraMatrix ComputationOperator TheoryComputational Linear AlgebraFundamental Theorem Of AlgebraSpectral Theorem
The Fundamental Theorem of Linear Algebra describes how an m×n matrix defines a linear transformation from ℝⁿ to ℝᵐ and characterizes solutions to Ax=b via four related subspaces. The paper’s purpose is to provide memorable visualizations of the matrix’s action on the four fundamental subspaces. The method visualizes Ax as a linear combination of A’s columns, shifting focus from numeric multiplication to subspace geometry and illustrating how solutions to Ax=b arise from combinations within the column space. The article focuses on the theorem and its accompanying illustrative figures.
This paper is about a theorem and the pictures that go with it. The theorem describes the action of an m by n matrix. The matrix A produces a linear transformation from Rn to Rm-but this picture by itself is too large. The truth about Ax = b is expressed in terms of four subspaces (two of Rn and two of Rm). The pictures aim to illustrate the action of A on those subspaces, in a way that students won't forget. The first step is to see Ax as a combination of the columns of A. Until then the multiplication Ax is just numbers. This step raises the viewpoint to subspaces. We see Ax in the column space. Solving Ax = b means finding all combinations of the columns that produce b in the column space:
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Roger Howe · American Mathematical Monthly · 1983 · 61 citations