Let a1, ..., a be n linearly independent vectors in R". Prove that if a vector b in R" is orthogonal to all the vectors a1,... , an, then b 0. For the following matrix, find the eigenvalues and also those eigenvectors that corre- spond to the real eigenvalues: 2 1 -1 0 1 2 0-2 Suppose A is a square matrix and let A be an eigenvalue of A. Prove that if det A # 0, then A 0. In this case show that 1/A is an eigenvalue of the inverse A-. Is given matrix diagonalizable? If so, what are the corresponding matrices P and D? 1 3 4 3 1 0 4 0 1 1 1 0 (b) 110 (c) 0 0 2

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.