I've made a anki deck based on the textbook "Linear Algebra" by Hoffman/Kunze. It was generated by Gemini, If there are any suggestions, improvements, or errors, please contact [email protected]
| Front | State the key lemma for proving simultaneous triangulation/diagonalization. |
| Back | Let \(\mathcal{F}\) be a commuting family of triangulable linear operators on \(V\). If \(W\) is a proper subspace of \(V\) which is invariant under \(\mathcal{F}\), then there exists a vector \(\alpha \in V\) such that \(\alpha \notin W\) and for each \(T \in \mathcal{F}\), the vector \(T\alpha\) is in the subspace spanned by \(\alpha\) and \(W\). (This means there is a common characteristic vector for the operators induced on the quotient space V/W). |
| Front | What are the properties of a projection E? |
| Back | A linear operator \(E\) is a projection if and only if \(E^2 = E\). That is, applying the projection twice is the same as applying it once. |
| Front | State Cramer's Rule for solving a system of linear equations (AX=Y). |
| Back | If (A) is an (n \times n) matrix with (\det A \neq 0), the unique solution to the system (AX=Y) is given by: (x_j = \frac{\det B_j}{\det A}) where (B_j) is the matrix obtained by replacing the j-th column of (A) with the column vector (Y). |