Use the inner product (f, 9) = f(2)g(=) dæ in the vector space P(R) of polynomials to find (f, 9), ||f||, ||9||, and the angle afg between f(x) and g(æ) for f(x) = 10a? – 3 and g(æ) = -9æ + 10. (f, 9) = ||F|| = I|g|| a fg

Q: Calculate F. dĩ, where C is the square of (1,1, 1), (–1, 1, 1), (–1, –1,1) and (1, –1,1), with…

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Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B'A) Given the…

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Q: Consider the vector space P, of polynomials with real coefficients of degree at most two together…

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Q: Determine whether the vectors u = 2 – x + 3x?, v = 4 +x + 2x?, and w = 8 - x + 8x² span P2 (the set…

A: The set {p1,p2,p3} will span p2 if and only if they are independent  so, here we test if the vectors…

Q: Given that xo= -1 and x1=1. Then the basis of the first Lagrange polynomial Lo(x) and L,(x) are…

A: Given that        x0 = -1 and x1= 1 On the basis of the first Lagrange polynomial we have to find…

Q: Use the inner product (p, a) = agbo + azb1 + azbz to find (p, a), P||. ||9||, and d(p, q) for the…

A: Given: Inner product p,q=a0b0+a1b1+a2b2

Q: Let x and y be vectors in the R³. Suppose that x is a scalar product of 2 and the vector y. Prove…

A: x and y are vector in R3, x is scalar or dot product of 2 and y vector,  it means x = 2.y,

Q: 2. Find the value of a for which the vector à = (axy – z³)î + (a – 2)x²j + (1 – a)xz²k have it's…

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Q: What is the dimension of the vector space S=abc detEM2x3(R): a=b=c=0}

A: Dimension of vector space is number of element in basis

Q: If T is defined by T(x) = Ax, find a vector x whose image under Tis b, and determine whether x is…

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Q: . u. ): Is the set S = {r +x-1,22 - r+3,-a+3z-2}) a base for the Pa(r) vector space which is a…

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Q: Suppose A is a 2x2 invertible Martix. Is the following statement true or false?  For each vector b,…

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Q: Let V=R^2 and let H be the subset of V of all points on the line 2x+3y=6. does H contain the zero…

A: Given  V=R2 and let H be the subset of V of all points on the line 2x+3y=6.

Q: Find the coordinate vector [p(x)]B of p(x) = x² + 4x + 4 with respect to the basis B = {x³ + x – 2,…

A: According to the given information, it is required to find the coordinate vector.

Q: Suppose that you have three vectors: f1 (x) = 1, f2 (x) = x – 1, and f3 (x) = } (x² – 4x + 2), that…

A: Given:fifj = ∫0∞fixfjxe-xdxf1x = 1f2x = x-1f3x = 12x2-4x+2D = ddx

Q: 3- The scalar component of the vector F in the direction of K is (K.F А- .K, В- K C-

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Q: Consider the set of vectors S = {(1, 0, 0), (0, 1, 0)} and the field F = R of real numbers, then…

A: The given set  of vectors S=1,0,0,0,1,0

Q: The set of all polynomials of degree 7 under the standard addition and scalar multiplication…

A: Given that the set of all polynomials of degree 7 under the operation of addition and multiplication…

Q: Sketch the vector field. y F(7,9) = ( - 55) 2' 2 -2 -2 -2 -2 -2 2. 2. 2. 2. 2.

A: The objective is to sketch the given vector field

Q: Consider the real vector space M2x3(R) with the usual inner product (А, В) — tr (B'A) Given the…

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Q: Consider the inner product on R2 defined by ((x, x2), (y, y2)) = x11 + 4x2y2. Which of the following…

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Q: Use the vectors a = [-1,0,1], b = [1,-1,5], and č= [0,-1,-2] to prove that a(bxč) # (åxb)xč

A: The answer is given below:

Q: Find the coordinate vector of Prelative to the basis S= {P1P2P3} P1 = 1+x +x², p2 = x +x, p3 = x:p =…

A: Given: p=6-x-9x2p1=1+x+x2p2=x+x2p3=x2

Q: Show that F = (3, 1, 2) is conservative. Then prove more generally that any constant vector field F…

A: Given: F = (3, 1, 2) To show: F is conservative. Also to show any constant vector field F = (a, b,…

Q: 21. Apply the conjugate-gradient method to the problem Ax = b, where 1 A = 1 1 %3D and b = 1 Verify…

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Q: 13. Recall that nonzero vectors u and are called collinear if there exists = a nonzero scalar a such…

A: Given data:- Nonzero vectors are u→ & v→ Nonzero scalar α v→=αu→

Q: Consider the vector space V = P over the field of real numbers F = R. Is the vector 4 – x – x 2 in…

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Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B'A) Given the…

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Q: Consider an inner product on two P₂ vectors as (f, g) = f(-1)g(-1) + f(0)g(0) + f(2)g(2): 1) (−4 +…

A: The inner product on two P2 vectors as f,g=f-1g-1+f0g0+f2g2

Q: Determine a scalar r such that Ax = rx, where

A: Matrix: A=12-11014-45x=-12141AX=12-11014-45.-12141

Q: If {v1, V2, V3} is an orthogonal set of nonzero vectors in an inner product space V, then prov that…

A: This question is related to linear algebra (inner product space).

Q: In the vector space of all real valued functions, find O (1+cos 2t, cost sint, cos² t) O (sin 2t, 1+…

A: Given spanning set is sin 2t, 1 + cos 2t, cos t sin t, cos2 t

Q: For (a, B) E R?, define a function (-, -) : R² × R? → R by setting arr' + xy + Yx' + Byy' . 1. Find…

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

A: Suppose  P7=f(x) | f(x) is a polynomial of degree 7. Then we have to check why this set is not a…

Q: Show that ℝ2 is a vector space with usual addition and multiplication.

A: To prove that R2 is vector space we have to prove following; 1. R2 is abelian group under addition.…

Q: Use the Wronskian to determine if the function vectors 7,= x? + e* and 72= e²x are linearly…

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Q: prove that 12 div A; = √√√x² {√8 8¹k Ax} = dxr div A¹, where A; and Ai are the covariant and…

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Q: : If v, v1, Vz are 3 vectors in a vector space V such that v – 9vi – 3v2 = 0. Then, v belongs to…

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Q: 3. 2 Find a non-zero vector in R* which is orthogonal to 1 and 2 1

A: Given vectors are 1234, 3121 and 2311. We have to find the non-zero vector in R4 which is orthogonal…

Q: Use the inner product (f, 9) = | $(2)g(x) dæ f(r)g(r) dr in the vector space C°[0, 1] of continuous…

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Q: Let 12, v, w be vectors In an inner product space such that //u|| =5, ||v||=5, and (u, v) =-2. Use…

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Q: Consider the vector space P, with the standard inner product. Calculate || p || and d(p,q) for the…

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Q: Find all vectors in R2 that are orthogonal to u = (2,1).

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Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B' A) Given the…

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Q: Find the divergence of the following vector field. F = = (5x2 – 5y²,7y² – 7z² ,8z² – 8x²) The…

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Q: 3. Use the Wronskian to determine whether the functions f1 = e*, f2 = sinx and f3 = x³ are linearly…

A: The Wronskian approach helps to find whether the given functions are linearly dependent or not. If…

Q: If Q: R" → R is a quadratic form, explain how to determine the maximum and minimum values that Q…

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Q: Show that (x, y) = 4x1y1-7x2Y2 for vectors x = (x1, x2) and y = (V1 Y2) in R2 defines an inner…

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Q: Q1. In each part determine whether the given vectors are orthogonal with respect to the Euclidean…

A: Two vectors u , v are orthogonal if and only if u.v=0 (a) Given u = 2 , 3 , -1 , 0v = (3 , 6 , -2 ,…