Linear Algebra Answers

Find an expression for a square matrix A satisfying A² = I_n, where I_n is the n × n identity matrix. Give 3 examples for the case n = 3.

Use Theorem 4.2.1 to determine which of the following are subspaces of P3. (a) All polynomials a0 + a1x + a2x2 + a3x3 for which a0 = 0. (b) All polynomials a0 + a1x + a2x2 + a3x3 for which a0 + a1 + a2 + a3 = 0. (c) All polynomials of the form a0 + a1x + a2x2 + a3x3 in which a0, a1, a2, and a3 are rational numbers. (d) All polynomials of the form a0 + a1x, where a0 and a1 are real numbers?

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Which of the following are subspaces of F (-infinity,+infinity)? (a) All functions f in F (-infinity,+infinity) for which f(0) = 0. (b) All functions f in F (-infinity,+infinity) for which f(0) = 1. (c) All functions f in F (-infinity,+infinity) for which f(−x) = f(x). (d) All polynomials of degree 2.

Assume T and S are matrices of the same size, Prove or disprove that (T+S)² is symmetric , skew symmetric or neither