Let P3(R) be the set of all polynomials of degree 3 or less with real coefficients. Prove that P3(R) is a vector space when considering + as point-wise sum and · as point-wise scalar multiplication (that is, if p1, P2 E P3(R), then pi +p2 is the polynomial such that p1 + P2(x) = p1(x) + p2(x), while a · p1 is the polynomial such that a · p1(x) = ap1(x)). Can you find a finite subset of P3(R) that spans P3 (R)?

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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.