Consider the bases B= {(3, -1), (1, 1)}, B'= {(2, 5), (1, 2)}, and S= {(1,0), (0,1)} of R'. If [u], =| find [u], and [u]s

Q: Find the coordinates of the vector v = (7,4) with respect to the basis U₁ = (2, 1) and u₂ = (3,1).

A: The answer is given below:

Q: Consider the vectors i = (2, – V3, 1) and ū = (-1, v3, 4). Find the orthogonal decomposition of u…

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Q: Find a basis for the plane 4 + 3y + 5z = 0 in R. %3D

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Q: Write down a basis of R 4 where you don't use any of the standard unit vectors (e1, ... , e4). Show…

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Q: Show that ?? = {(1,2,3), (−1, −1,1), (5, −4,1)} is an orthogonal basis of ℝ3, and use the Fourier…

A: Definition: A subset of a vector space V is linearly independent, if  Theorem: If is a orthogonal…

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A: The basis in n-dimensional space is called the ordered system of  linearly independent vectors. Let…

Q: Consider R3 with the inner product defined by hu, vi = 3u1v1 + 2u2v2 + u3v3, for u, v ∈ R3. a.)…

A: Consider ℝ3 with inner product defined by u,v=3u1v1+2u2v2+u3v3  for u, v∈ℝ3  and the basis…

Q: Prove that if {y1,...,yn} is an orthogonal set of nonzero vectors, then the vectors {x1,...,xn}…

A: Let consider the orthonormal set of nonzero vector be {w1, w2 …wn} instead of {y1...,yn} and the…

Q: (i, 2, 3i), v = Co over C. Are these vectors linearly independent? Can they form a basis 5.…

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Q: the vectors u1 = (2,−3,1), u2 = (1,4, −2), u3 = (−8,12, −4), u4 = (1,37, −17), and u5 = (−3,−5,8)…

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Q: Find the coordinate vector[x],of x relative to the given basis B= {bl,., bn}. --]--[}]--[1] bị = ,b2

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Q: Find a basis of R3 that contains the vectors (1,0,1) and (1,-1,1)

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Q: It can be shown that the vectors u1 = (2,−3, 1), u2 = (1, 4,−2), u3 = (-8, 12,−4), u4 = (1, 37,−17),…

A: A basis is a subset of a vector space that spans the vector space and is linearly independent.

Q: If vi, v2, ..., Vn are independent vectors in R", then they must form a basis of R". True False…

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Q: Verify that the polynomials 2 – x², x' – x, 2 – 3x² and 3 – x' form a basis for P3(x). Express cach…

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Q: Show that fu,, u,) is an orthogonal basis for R?. Then express x as a linear combination of the u's.…

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Q: Find a basis fo

A: Given,            A=3-916-26-161-38

Q: Given the bas is x, cwhere X= X,J for R3 2. 3. use the Gram - Schmidt process to transform the basie…

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Q: Find the coordinate vector of p(x) = 3 - x + 4x2 with respect to the basis B = {1, 1 + x, -1 + x2}…

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Q: Let uj = (1,1,2), uz = (1, 3, –2), Uz = (4, –2, –1) in R’. Show that u , U2 , Uz are orthogonal, and…

A: u1=112,  u2=13-2,  u3 =4-2-1 vectors are orthogonal if their dot product is zero  . u1·u2=11213-2…

Q: Suppose {u, } form an orthogonal basis in R2 and y = 2ū - 7 where u = (8) 6 If W = Span{}, what is…

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Q: Show that the vectors (1, 1, 2), (1,2,5), (5,3,4) of R'(R do not form a basis of R'(R).

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Q: Find the coordinate vector of p(x) in P., relative to the basis S= {p,.P.-P;}, where p(x) =…

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Q: Suppose that a₁,a2,..., a, form a basis of R" and y = ₁²₁ +2₂₂ + ... + n Anan with 2; = 0. Prove…

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Q: In the vector space P,, of polynomials of degree at most two, the coordinates of the vector P(x)…

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Q: and ŷ = Find the values of y, and y, such that vectors = constitute an orthonorma L0.8337! basis in…

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Q: 4. Let {V1, ... , Vn} be an orthogonal basis of R". For any vectors x, y in R" show that Σ (х - V,)…

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Q: Define a basis v1, v2 of R² by vị obtained by applying the Gram-Schmidt Procedure to v1, v2? 2 by…

A: Consider the provided question, A Basis of a vector space. Let V be a subspace of Rn for some n. A…

Q: Find the coordinate vector of = -1 with respect to the basis B = for IR3. 2 B

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Q: Let uj = (1,1,2), uz = (1,3, –2), uz = (4, –2, –1) in R’. Show that uj , uz, Uz are orthogonal, and…

A: The answer is given in this below photo

Q: Consider an inner product on two P3 vectors as (f, g) = f(-1)g(-1) + f(0)g(0) + f(1)g(1) + f(2)g(2):…

A: Since you have posted multiple subpart question, we will solve the first three sub parts for you. If…

Q: 3) Let B1 = {u1, U2} and B2 = {v1, v2} be the bases for R in which u1 = (2, 3), vị = (1,3) and v2 =…

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Q: Suppose that P2 has an inner product defined on it in which the set {1, x, 1 +x+x} is an or-…

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Q: Use Gram-Schmidt to find an orthogonal basis for R' if B={(2, -5), (1, 3)}.

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Q: 2. Find a basis for R that includes the vectors and 2

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Q: Let {v1, ... , vn} be an orthogonal basis for Rn and let W = span(v1, ... , vk)· Is it necessarily…

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Q: Find a basis for R which contains as many vectors as possible of the following quantity: {(1, 2, 0,…

A: Let us first verify whether the above vectors are linearly independent or not. Thus, the vectors…

Q: The Gram-Schmidt process transforms the basis u=(1,2), u2=(2,-6) of R into the orthogonal basis…

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Q: Find a basis for the plane 4x + 3y + 5z = 0 in R³.

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Q: In the vector space P, of polynomials of degree at most two, the coordinates of the vector…

A: The basis of the given polynomial is: B= e1=1+2x,  e2=x-3x2,  e3=1+x2

Q: ´) Consider the vectors - - (-) · · - () ₁ - - () a= 2 Do these vectors form a basis of R³? If yes,…

A: They form a basis for R3. Explanation is as follows

Q: Show that the vectors (1, 1, 2), (1, 2, 5), (5, 3, 4) of R'(R) do not form a basis of R'(R).

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Q: a) Find , \pl, d (p, q), and then find the angle between them for the polynomials in P2 if p(x)…

A: a) given polynomials belong to P2 , so their basis set will be {1,x,x2}  p(x)=-3x2+2x-5 =(-5,2,-3)…

Q: suppose that x↓1,..,x↓(n-1) are linearly independent vectors in R^n and also x↓n is notin the span…

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Q: Let x, y, and z be vectors in Cn and let α and β becomplex scalars. Show thatz, αx + βy} = α z, x} +…

A: Given: z,αx+βy=αz,x+βz,y

Q: Suppose that P2 has an inner product defined on it in which the set {1, r, 1 +x + x²} is an or-…

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Q: show that a set of vectors{et cos(2t), e¯t sin(2t)} form a basis of solution space for x" + 2x' + 5x…

A: Substitute the value of x in the solution space and see if the value satisfies the equation, if the…

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.