Let x = (x1, ... ,xn) and y = (y1,...,yn) ∈ Rn be fixed. Recall that Rn is a vector space, and that it is an inner product space when equipped with ⟨x,y⟩ = Σ nj=1 xnyn, where Σ is the summation notation.  This problem walks you through in proving the Cauchy-Schwartz inequality for this inner product. (a) For z ∈ R, Verify the following identity:P(z) = (x1z + y1)2 + (x2z + y2)2 + ··· + (xnz + yn)2          = (Σnj=1 x2j )z2 + 2 (Σnj=1? xjyj)z + Σnj=1 y2j. (b)  Explain why this polynomial P(z) is always nonnegative.

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A: The complete solutions are given below

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.