(5) Let Pe B(H) be a nonzero idempotent. Show that the follow- ing are equivalent. (a) P is a projection. (b) |P|| = 1. (c) P is normal.

Prove that Z[i] is a Euclidean domain.

Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1,y1)(x2,y2) = (x1 + x2 − 2, y1 + y2) c(x,y) = = (cx - 2c+2, cy – 5 c + 5). (a) Show that 1 is a scalar multiplication identity, that is: 10(x,y) = (x,y). (b) Explain why V nonetheless is not a vector space. Hint: Check for if scalar multiplication distributes over vector addition.

Which of the following are vector spaces? Justify your answer. (a) The set of all polynomials of degree 3. (b) The set of all vectors x = (x1, x2, x3), satisfying 3x₁ + 5x2 − 9x3 = 2023. (c) The set of all vectors x = (x1, x2, x3), satisfying 2024x₁ + x2 = 0 has a unique solution. (d) The set of all 3 × 3 matrices such that Ax= (e) The set of all n × n (n = N) diagonal matrices. - x3 = 0. -