2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider B = {1+x = 2a°, –1+ x + a*, 1 – x + a²} , B' = {1- 3x + a², 1 – 3x – 20², 1 – 2x + 3a²}. (a) Show that B and B' are bases of P2(R). (b) Find the coordinate matrices of p(r) = 9x² + 4x – 2 relative to the bases B and B'. (c) Find the transition matrix Pg→B'. (d) Verify that [x(p)]B = PB-¬B' [x(p)B].

Q: Prove that the vectors (1. 1. 0) (1 2 3) and (2. –1 5) form a basis for R3

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Q: Let 01, 02, 03 be three nonzero vectors in a vector space V. Suppose that V1, V2, V3 are orthogonal…

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Q: Let V=R². For (u₁, U₂), (v₁, v₂) € V and a € R define vector addition by (u₁, ₂) (V₁, V₂) = (U₁ +…

A: The given vector space is .For any  and , the vector addition and scalar multiplications are defined…

Q: Let V₂ = {(a, b)|a, b = R} with the following operations: (a₁, α₂) + (b₁,b₂) = (a₁ + b₁ − 1, a₂ + b₂…

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Q: b) Let Pn be a vector space consisting of polynomials of at most degree n. The standard basis for P,…

A: value of k is 0 see calculation below

Q: Let B = (1, x, x², x³) be an ordered basis for P3. Find the coordinate vector 8x³ + 2x²7x - 9…

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Q: (3) Let V be R², the set of all ordered pairs (x, y) of real numbers. Define an operation of…

A: Given, is the set of all ordered pairs  of real numbers.The operation of addition is defined by,…

Q: Find scalars x, y and z such that x(2, 1, 4) + y(1,−1, 3) + z(3, 2, 5) = (6,11,6)

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Q: Let x 0 and y 0. (a) If xly, show that {x, y} is a linearly independ- ent set. (b) Extend the result…

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Q: For the vectors à = (1, 0, −4) and ¯` = (5, 2, 1) what is the scalar a projection of onto a 1/17 √17…

A: The vectors are given by÷

Q: Find the limit. Enter DNE if the limit does not exist. lim (2,y) + (0,0) x² + 14y2

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Q: 4. Find the scalar and vector projections of b onto a (a) a =, b = (b) a =, 6=

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Q: Represent the vector ⟨16,8,−14⟩ in terms of the ordered basis B={⟨5,−2,−2⟩,⟨1,−4,2⟩,⟨−1,−5,3⟩}.

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Q: 7) Find all vectors v in F such that 2v + ) -)

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Q: Find an orthonormal basis of the plane x1+ 5x2 – x3 = 0. }

A: Introduction: A set of vectors that is orthogonal to each other and spans a plane is the set of…

Q: nd scalars a, b and c such that a(2,0,1) +b (1,3,2)+c(1,0,1)=(7,3,4)

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Q: 2. The vector x = vector [x]B A 1 3 0 = [1] a [9] in R2 relative to the orthogonal basis B = What is…

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Q: (au1, 2). respectivel Let V = {(r,y)|r,y E R}, with addition and scalar multiplication defined as u…

A: Here, V=(x,y)|x,y∈ℝ with addition and scalar multiplication are defined as…

Q: "On R², define the operations of addition and scalar multiplication as follows: (x₁, x₂)+(V₁ Y2) =…

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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.