[IDEA] Linear Algebra - Rowspace(A) = Colspace(A^T)

[zakuArbor]

• not intuitive
• https://www.symbolab.com/solver/step-by-step/gaussian%20elimination%20%5Cbegin%7Bpmatrix%7D1%265%268%26-8%26-1%5C%5C%20%20%20%201%266%268%26-5%260%5C%5C%20%20%20%202%2615%2616%26-1%263%5C%5C%20%20%20%200%266%262%268%266%5C%5C%20%20%20%201%2644%2622%2639%2638%5C%5C%20%20%20%20-1%26-3%26-7%2616%26-5%5Cend%7Bpmatrix%7D?or=input

v.s.

https://www.symbolab.com/solver/step-by-step/gaussian%20elimination%20%5Cbegin%7Bpmatrix%7D1%261%262%260%261%26-1%5C%5C%20%205%266%2615%266%2644%26-3%5C%5C%20%208%268%2616%262%2622%26-7%5C%5C%20%20-8%26-5%26-1%268%2639%2616%5C%5C%20%20-1%260%263%266%2638%26-5%5Cend%7Bpmatrix%7D?or=input

• the idea is that you have the colspace of the transpose but it's not reduced but the rowspace is reduced. [rowspace(A) | colspace(A^T)] is probably what you need to do and the reverse to show they are a subset of each other so that they are equal
• use a simple example - the one I linked is too complex

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• recall rowspace preserves row operations but not dependencies but swapping rows and etc makes it unadvised to get rowspace from matrix but instead should get from RREF
• colspace preserves linear dependency but not row operations