a) Let P be the set of all polynomials f(x), and let Q be the subset of P consisting of all polynomials f(x) so f(0) = f(1) = 0. Show that Q is a subspace of P. b) Let P be the set of all degree ≤ 4 polynomials in one variable x with real coefficients. Let Q be the subset of P consisting of all odd polynomials, i.e. all polynomials f(x) so f(−x) = −f(x). Show that Q is a subspace of P. Choose a basis for Q. Extend this basis for Q to a basis for P.

Q: 1. Consider the set V of all polynomials of degree two or less, along with the following definitions…

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Q: a) range 1 0 0 11 00 10 b) range 1 1 01 1 c) span ((1,-1, 1), (−1, 1, 1), (1, 1, −1), (1, 1, 1)) d)…

A: Note : '' As per guidelines we will solve first three subparts. If you want any specific subpart to…

Q: 7.4.3 Let X = {1,2,3}. Define the relation R = {(1, 1), (1,2), (1,3), (2, 1), (2, 2), (3, 1), (3,3)}…

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Q: 6) Let R1 and R2 be relations on a set A such that R1 = {(1, 2), (2, 1), (3, 2), (3, 3)} R2 = {((1,…

A: Solution

Q: Determine if 0 >=(61-6-J-RA) is in the span of = 0

A: Given: -1225 To check:If -1225 is in the span of Φ=1100, 1001, 0110, 0011 or not.

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Q: Prove that V is a vector space with the operations

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Q: b) A function g: Z → Zis defined by the rule g(m, n) = m + n Determine whether it is onto or not..

A: We will answer the first question as we don't answer multiple questions at a time. Please resubmit…

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Q: 20 - If f is a function on a set A= {1,2,3,4,5} such that f= { (1,2), (2,3), (3,4), (4,x), (5,5)}.…

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Q: A function, F: R × R ---› R x R has been defined as follows: F(x, y) = (3y - 1, 1 - x) for all (x,…

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Q: Let A = {1, 2, 3}. Give an example of a relation R on A such that Ris symmetric but is not…

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Q: Find the minimal polynomial for  ∛2 + ∛4  over Q.

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Q: What is the kernel and image of the map

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Q: Assume (r1, y1) ~ (x2, Y2) → f(x1,Y1) = f(x2, Y2). Prove that ~ an equivalence relation.

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Q: Let G = (-1,1) and (x,y) --> x ○ y = x + y/1 + xy is a binary operation on G. Let R+ = (0, ∞), with…

A: Given: G,O  with G=(-1, 1) ,  (x, y)→ x O y = x+y1+xy   and ℝ+, ·, ℝ+ = (0, ∞) with standard…

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Q: Prove that gcd(a, b) is the least natural number representable in the form ax + by with x, y ∈ Z.

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Q: Let R be a relation on the set of all functions from Z to Z defined by: R = {(f,9)|f(1) = g(3) or…

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Q: Let R be the relation on Z given by R= {(a,b) : a < b}. %3D (a) Find R-1. (b) Find R which is Z x Z…

A: relation on Z.

Q: Prove that the function f(x1,x2) = In (e*i+= + eV=i+1) is convex over R².

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Q: the book is "First Course on Fuzzy Theory and Applications"

A: To show: The given function satisfies the condition

Q: Let such that 1 0 1 -1 2 -6⠀⠀⠀-0-0 1 1 3 -1 b 000 0 Then there exist scalars b A True False = X1,…

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Q: Consider relation R={(x,y)|x divides y,xEA,y€A} on the set A={2,3,4,5,10,12,25,30}. (a) Check…

A: We are given a relation : R={(x,y)|x divides y,x∈A,y∈A}  on a set A={2,3,4,5,10,12,25,30}

Q: let G be the set of all triples of the form (k1, k2,1) or (k1,k2,-1), where the ki, i=1,2 are…

A: Given :  G = k1, k2, ±1 : k1, k2 ∈ℤ with the operation  k1, k2, 1l1, l2, -1 = k1+l2, k2+l1, -1 To…

Q: Which functions are one-to-one? Which functions are onto? Describe the inverse function for any…

A: One to one: A function f : A → B is one-to-one if for each b ∈ B there is at most one a ∈ A with…

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.