Let P₂[R]=(a+a₁x+aα₂x² | Q₁ER,I=0,1,2] =Set of all real polynomials in x of degree ≤2 The vector addition is defined by (a + a₁x+a₂x)+(b₂+ b₁x+b₂x²)=(a+b)+(a,+b)x+(a₂+ b₂)x where a, b, ER, i = 0,1,2. Verify all properties of vector addition for real vector space P₂[R].
Question
Let P₂[R]=(a+a₁x+aα₂x² | Q₁ER,I=0,1,2] =Set of all real polynomials in x of degree ≤2
The vector addition is defined by (a + a₁x+a₂x)+(b₂+ b₁x+b₂x²)=(a+b)+(a,+b)x+(a₂+ b₂)x where a, b, ER, i = 0,1,2.
Verify all properties of vector addition for real vector space P₂[R].
Solution
To verify the properties of vector addition for the real vector space P₂[R], we need to check the following properties:
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Closure under addition: If we add two vectors in P₂[R], the result should also be in P₂[R]. Given two vectors p(x) = a + a₁x + a₂x² and q(x) = b + b₁x + b₂x², their sum is (a+b) + (a₁+b₁)x + (a₂+b₂)x², which is also a polynomial of degree ≤2. So, P₂[R] is closed under addition.
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Commutativity: The order of addition should not matter. For any two vectors p(x) and q(x) in P₂[R], p(x) + q(x) = q(x) + p(x). This is true because (a+b) + (a₁+b₁)x + (a₂+b₂)x² = (b+a) + (b₁+a₁)x + (b₂+a₂)x².
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Associativity: For any three vectors p(x), q(x), and r(x) in P₂[R], (p(x) + q(x)) + r(x) = p(x) + (q(x) + r(x)). This is true because addition of real numbers is associative.
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Existence of additive identity: There should exist a zero vector in P₂[R] such that adding it to any vector in P₂[R] does not change the vector. The zero vector in P₂[R] is 0 + 0x + 0x². For any vector p(x) = a + a₁x + a₂x² in P₂[R], p(x) + 0 = a + a₁x + a₂x² = p(x).
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Existence of additive inverse: For every vector in P₂[R], there should exist another vector in P₂[R] such that their sum is the zero vector. For any vector p(x) = a + a₁x + a₂x² in P₂[R], its additive inverse is -a - a₁x - a₂x². Adding these two vectors gives the zero vector.
Therefore, all properties of vector addition for the real vector space P₂[R] are verified.
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