Let P2 be the vector space of all real polynomials of degree at most 2. Let p1, P2, P3 E P2 be given by p1 (x) = 2, p2(x) = x + 2x², and p3(x) = 6x + aa². a) (You are free to assume that the functions 1, x and x? are linearly independent.) b) combination of p1, P2 Find the condition on a e R that ensures that {P1, P2, P3} is a basis for P2. In the case that a = 6, write the function p(x) = 2 – x + x2 as a linear and P3.

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A: Let P3 be the vector space of polynomial functions of degree less than or equal to 3 over the field…

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A: Note:  As you requested only Part C(I) so I answered here only that.

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Q: The set of all polynomials of degree 6 under the standard addition and scalar multiplication…

A: The answer is given below

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Q: The set of all polynomials of degree 6 under the standard addition and scalar multiplication…

A: It is not closed under addition.

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Q: 6. Determine the basis for the null space of -2x + 2y + 2z] a) 3x + 5y + z 2y + z х — 2у — z b) -x +…

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A: Let's find.

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Q: (b) Show that the set of all positive real numbers, with x+y and cx redefined to equal the usual xy…

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A: 4. Is false and others are true i have justified the answer see the step 2

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Q: 6. Determine the basis for the null space of: -2x + 2y + 2z] Зх + 5у + z 2y + z a) х — 2у — z b) -x…

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Q: 6. Determine the basis for the null space of: -2x +2y + 2z] Зх + 5у + z 2у +z a) х — 2у — z b) —х +…

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Q: x=y=z

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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.