' Define an inner product on the vector space P2(t) as following 3 (p(), g(t) -Σ α.)g (a,) i=0 where ao, a1, a2, a3 = -3, –1, 1,3 Construct the orthogonal bases from p(t) = 1, q(t) = t,r(t) = t? space P2(t).

Q: c) Show that the vector field W(x, y, z) = (2yz+ xz – x², –2xz – yz + y?, y² – x² + z²) - - is a…

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Q: 2. Let a, B, y be three vectors in the inner product space V½ (R) with the standard inner product…

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Q: 7 Given the inner product space (R³(R),+,·,· ), where xoy = 3x₁y₁ + 2(x₂Y₁ + X₁Y₂) + 3x₂Y/₂ + 2x3V3,…

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Q: Compute the vector assigned to the points P = (1, 1) and Q = (-3, -3) by the vector field F = (-y,…

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Q: Set V = {(x, y) | x ER, YER}. Define addition of vectors by the rule (x, y) + (x, y) = (y + y',x+x')…

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Q: Let v = [4 2 -1]" Considering the vector space R³ provided with the inner product defined by: (a, b)…

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Q: 6. Let V be a vector space over F and let u, v, we V, a, Be F. Which of the following is not a…

A: In this question, we will discuss about the properties of vector space.

Q: 2. Let V be a vector space of dimension n and let B = {v1,..., Un} be a basis of V. (a) Prove that…

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Q: Q2) (a) Use the definition = ]f(t)g(t)dt in T0,1] to find the components of f(t)=2+t in the…

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Q: V is a vector space with bases B = (v1, V2, v3) and B' = (w1, w2, w3). The change of basis matrix 1…

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Q: Given the operator S on the vector space R₁ [x] given by S (a+bx) = -a+b+ (at 2b) x A) What is SC1)…

A: We know that Pn is the vector space of all the polynomials of degree n+1 and the basis for this…

Q: 2. Use abbreviated index notation to prove or disprove the following statements involving arbitrary…

A: Since there are multipart, so here find the solutions of first and second part.

Q: Is the following set of vectors S = {v,, V2. V3} basis for R³ space? Where, a. v = (3,1,–4), v2 =…

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Q: 13. For the vector fields: A = x + 2y ŷ + 3z 2; B = 3y - 2x ŷ. Check both product rule given in…

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Q: - {{:) ()(}} (3) 2 B be an ordered basis of the vector space R³. Let the vector v = Find the…

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Q: Consider the vector space R" over R with the usual operations. (a) Show that 1 1 e1 e2 en and 1 1 1…

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Q: HIf vectors x, y, z, u in the real vector space p (the polynomials) are defined by x(t)=1, y (t)=t,…

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Q: 1 ²+1 (a) Find the inner product (f. 9) = (b) Verify the Cauchy-Schwarz Inequality. 2. Let f(x)= x…

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Q: : R^3-->R^3,T(x, y, z) = ( x + y, y + z, x - z)  (a)the basis of  the range(T) (b)determine if T…

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Q: If are arbitrary vectors in R2x2, then the mapping A = (A, B) || A|| = ||B| = [a11 a12] a21 a22 A =…

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Q: FTTT 10 5 10- -5- -10- X TTTTI 1C d.

A: We draw the sketch of vector field with the help of direction and length of vectors .

Q: 3) Let (v,, vy, v) be a set of vectors from a vector space R', then: a) The vectors are linearly…

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Q: Verify all vector space axioms under the given operations (x,y,z)+(x',y'z') = (x+x'+1,y+y'+2,z+z'+3)…

A: The objective of the question is to verify if the given operations satisfy all the axioms of a…

Q: Determine whether the following vectors in R' are linearly independent lineariy dependent: (i)…

A: As per our guidelines I am supposed to solve only one question. So kindly upload the other questions…

Q: Show that the vector space P, has the basis S = {1, x, x², x³}. %D

A: We have to solve given problem:

Q: 10) (6 points) Using the inner product (v, w) = 3v¡W1+5v2W2 with the vector space R?. a) Find a…

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Q: Let v1 = (1,2,1), v2 = (2,9,0), v3 = (3,3,4) . Show first that S= {v1, v2, v3} is a basis for ℝ^3.…

A: A linear map is a mapping A to B between two vector spaces that preserves the operations of vector…

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.