05-02-03 - Lecture 33 Andrei Antonenko May 2, 2003 1 Direct...

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Lecture 33 Andrei Antonenko May 2, 2003 1 Direct sum of vector spaces In the first part of this lecture we will consider some more concepts from the theory of vector spaces. Definition 1.1. Sum of two vector spaces U and V U + V is a vector space which consists of all vectors u + v , where u U and v V . Definition 1.2. Vector spaces U 1 ,U 2 ,...,U n are called linearly independent if from u 1 + ··· + u n = 0 , where u i U i it follows that u i = 0 for all i . Sum of the linearly independent vector spaces is called a direct sum of these vector spaces and is denoted by U 1 ⊕ ··· ⊕ U n . Definition 1.3. The vector space V is said to be equal to a direct sum of vector spaces U 1 ,...,U n V = U 1 ⊕ ··· ⊕ U n if any vector v from V can be represented as v = u 1 + ··· + u n , u i U i uniquely. For example, the plane R 2 is equal to a direct sum of x - and y - axes. Example 1.4. The space of all matrices is equal to a direct sum of the space of all symmetric matrices and all skewsymmetric matrices, since any matrix A can be uniquely represented as A = A + A > 2 + A - A > 2 , and one can check that A + A > 2 is always symmetric, and A - A > 2 is always skewsymmetric. More- over, the sum is direct, since if a matrix is both symmetric and skewsymmetric, it is equal to 0 -matrix. 1
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2 Invariant spaces Definition 2.1. Let A be an operator in vector space V . The subspace U V is called an invariant subspace with respect to operator A if A U U. This definition means, that the vectors from invariant subspace remain in this subspace after application of the operator A . Example 2.2. Considering the operator of rotation in the 3-dimensional space around some axes, we can see, that all the planes, perpendicular to the axes of rotation are invariant. More- over, the axes of rotation is invariant itself. If the basis { e 1 ,...e n } of V is such that first k vectors { e 1 ,...,e k } is a basis of U , then the matrix of the operator A in this basis has the following form: ˆ B D 0 C ! . Moreover, if the space V is equal to a direct sum of two subspaces V = U W , and { e 1 ,...,e k } is a basis of U , and { e k +1 ,...,e n } is a basis of W , then the matrix of A has the following form: ˆ B 0 0 C ! . Example 2.3. Consider the rotation in the 3-dimensional space about some axes be an angle α . In the basis { e 1 ,e 2 ,e 3 } , if the vector e 3 is directed along the axes of rotation, the matrix of this operator has the following form: cos α - sin α 0 sin α cos α 0 0 0 1 This matrix is consistent with the decomposition of R 3 into a direct sum of two invariant
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05-02-03 - Lecture 33 Andrei Antonenko May 2, 2003 1 Direct...

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