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Unformatted text preview: Lecture XXXIV Subspaces In the previous lecture we have seen that there are two methods for finding the inverse of a square matrix A . In the first method, we use the fact that if A is non-singular, then the row-reduced form of [ A : I ] is [ I : A 1 ]. In the second method, we take the matrix of cofactors A , consider its transpose ( A ) t , and 1 1 multiply it by detA to obtain the inverse of A : det A ( A ) t . The folowing theorem is an important determinant property. Theorem 1 Let A and B be two square matrices of the same size. Then AB | = . | | A || B | Definition 1 A set V is a vector subspace of R n if it satisfies the following properties: (i) 0 is in V . x , x + (ii) For any y in V , y is in V . x in V and any scalar c in R , c (iii) For any x is in V . Clearly, the set containing only the origin is a vector subspace. In R 3 , a vector subspace must be either the set previously mentioned, a line through the origin, a plane through the origin, or all of R 3 . ....
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- Two '04