When a matrix cannot be inverted, how do you solve for the best fit?
In short: Linear algebra often relies on finding inverses, but what happens when a matrix is not invertible? This video explores the four fundamental subspaces and introduces the pseudoinverse as a powerful tool for finding the best possible solution in least squares problems.
The four fundamental subspaces consist of the column spaces and nullspaces of a matrix A and its transpose A^T. These subspaces exist in m-dimensional and n-dimensional spaces, providing a geometric framework for understanding linear systems. When m equals n and both nullspaces contain only the zero vector, the matrix is invertible.
However, when A is not invertible, the standard inverse A^-1 does not exist. In these cases, the goal shifts to finding a vector x within the row space that minimizes the distance || Ax - b || within the column space. This approach, known as least squares, utilizes the pseudoinverse to identify the optimal vector when a perfect solution is unreachable.
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- Summary by Intellect · source: The Four Fundamental Subspaces and Least Squares
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