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**Unformatted text preview: **Topics Covered by Midterm 1: (1) Gauss elimination method for solving systems of linear equa- tions. It is used everywhere where you actually need to solve something, and not just wave hands... (2) Notions: Matrices of coefficients of linear systems: rank. Linear spaces and linear independence of vectors. (3) Matrices: Addition, multiplication (be aware that AB negationslash = BA in general!). (4) Determinant (remember, defined only for square matrices!), Connection with rank, and therefore, with Gauss method. (5) Inverse of a (square only!) matrix. Two methods to find an inverse: actually find it (Gauss-Jordan) or to prove something about it (cofactors). A − 1 exists if and only if det A negationslash = 0. (6) Scalar product: Orthogonal, symmetric, skew-symmetric ma- trices and their complex analogues. Why it is easier to find x i such that x 1 −→ e 1 + ... + x n −→ e n = −→ f if ( e i , e i ) = 1 and ( e i , e j ) = 0 for i negationslash = j ?...

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