1. Let {V1,... , Vn} be an orthogonal basis of R". For any vectors x, y in R" show that (x . V;) (y V;) 3= X • y = i=1

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: If {u, v} is a basis for a vector space V, show that {u – 2v, au} is also a basis for V for every…

A: Here we use the definition of basis and linear combination of vectors as follows

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

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Q: Show that {u,, u2, uz) is an orthogonal basis for R. Then express x as a linear combination of the…

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Q: Find the coordinate vector of p (x) = 1 + 2x + 3x2   with respect to the basis ẞ = {l + x, 1 -x, x2}…

A: Here given

Q: Find a basis for null(T), where T : R^3 → R^2 is defined by T (x, y, z) = (−7x + 2y − 5z, 4x + 9y +…

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Q: Prove that the components of a vector x in a given basis {ê¿} are unique. That is, show that if you…

A: We will prove the given statement.

Q: Let (vi..... vn) be an orthogonal basis of R". For any vectors x, y in R" show that (x- vi) (y vi)…

A: Given v1,v2,…,vn be an orthogonal basis of ℝn. Let x,y∈ℝn. Since v1,v2,…,vn is basis, therefore:…

Q: 1 Let u₁= = U₂ = U3 = 1. and y 3 3 1 (a) Show that {u₁, U2, U3} is an orthogonal basis for R³. (b)…

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Q: 4. (a) Let L be the line in R' spanned by the vector (1,1, 1, 1). Find a basis for the orthogonal…

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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: Answer the following questions about the Gram-Schmidt process for obtain- ing an orthogonal basis…

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Q: (11) Find the dual basis {V4.V } of the stan dard basis of IR. Let T:IR → IR defined by T(xiy, z)…

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Q: 4. Let B = {V1, V2, Vn} be a basis of a vector space, V. ... a) Prove just ONE of the following: •…

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Q: Consider the basis B of R consisting of vectors and Find x in R whose coordinate vector relative to…

A: Basis is defined as the linearly independent set that span a vector space V. Any vector of V can be…

Q: Explain how fixing a basis v1, corresponding function in n variables of the form q(x1, · ·· , Xn) =…

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Q: Let W be the set of all vectors with x and y real. Find a basis of W-

A: Here, W is a set of vectors of the form xyx+y=x101+y011 with x and y are real numbers. Now, the…

Q: Let {v1,…,vn}{v1,…,vn} be an orthogonal basis of RnRn. For any vectors x,yx,y in RnRn show that…

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Q: Let W be the set of all vectors with x and y real. Find a basis of W1 x + y,

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Q: {( a + b+c+d b – c+ 2d а + 2с — d H = : a, b, c, d e R } C R³, and consider --(}) u = 1 (а) Is u E…

A: Given, H=a+b+c+db-c+2da+2c-d: a,b,c,d∈ℝ⊆ℝ3      ⋯⋯(1) a) Given, u=111 Consider the system,…

Q: Let u and v be two vectors in R°. Determine the most general condition under which {u, v, u × v}…

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Q: 9. "Find two bases in the vector space R* (over R) such that the only vectors common to both are…

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Q: 1 Find the vector projection of u onto v. u= 3i- 5j ++ 2k v=7i+j- 2k

A: Please refer the attached image for complete solution.

Q: b: Construct the orthogonal basis of set T= {v1,v2, V3} for R3 Where v = 19 v2 = 7 V3 = |1 Express…

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Q: 6. Let B= {(v,,V½, v½ } be a basis for R'. Use the Gram-Schmidt process to find an orthogonal basis.…

A: We need to find the orthogonal basis of R3 . The basis here is {v1 , v2 ,v3 } and we do not need to…

Q: 3 2 1 Let u [-]--[-]-[] -3 V 2 and w= 1 0 -1 4 (a) Show that B = {u, v, w} is an orthogonal basis.

A: The given vectors are:  u=3-30, v=22-1, w=114. (a) To show: {u,v,w} is an orthogonal basis. (b) To…

Q: The following vectors u,, u,, and u, form an orthonormal basis for R³. (금이) u, = (0, –1, 0), u2 = 0,…

A: We find the required linear combination

Q: 22. Find the vector x (with respect to the standard basis) determined by the coordinate vector (x]B…

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Q: Expand x = (a, b, c, d) with respect to the orthogonal basis of: 2 -1 1 Vị = V2 -2 3 Express your…

A: Note: Observe that the given orthogonal basis spans only two dimension subspace of ℝ4, therefore the…

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: Prove that a linearly independent system of vectors v1, V2, . .. , Vn in a vector V is a basis if…

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Q: 11. Find a basis for the set of vectors in R³ in the plane x + 2y + z = 0. [Hint: Think of the…

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Q: Find x with respect to the standard basis corresponding to the coordinate vector [x]g and basis B. 2…

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Q: Show that B = {v1, v2, v3} is an orthogonal basis for R3: v1 = (3,−3,0), v2 = (2,2,−1), v3 =…

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

A: We have to solve given problem:

Q: 17. Let u= (1,-1,0) and v= (1,2,1). (i) Verify that u and v are linearly independent. (ii) Extend…

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Q: (a) Show that {v1, v2, V3} are linearly independent, where 1 V1 = 1 and v2 = and v3 = -2 Hence {v1,…

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Q: 10. Consider the plane 2x - 3y+4z 0. Find a basis B of this plane such that (B = for ū =

A: Solution:-

Q: Let {u1, uz U3, U4, U5} be an orthogonal basis for R$, y a vector in R$, W1 = Span{u1, uz} and W2 =…

A: To find - Let u1, u2, u3, u4, u5 be an orthogonal basis for R5, y a vector in R5, W1 = Spanu1, u2…

Q: 1 Q.2) How can you show that the set{ 2 2} form basis of xy- plane in R3? 3

A: GIVEN SETS : The given sets are : 120,123 The augmented matrix is : 120123R2→R2-R1~120003

Q: The Gram-Schmidt process transforms the basis u=(1, 2), u2 =(6, 2) of R2 into the orthogonal basis…

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Q: - --- Let B = Is Ba basis ofR ? Justify and x = %3D 2 3 3 your answer. If yes, compute xR [x]

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Q: 4. Let { u1, Uz, U3, U4, U5} be an orthogonal basis for R$, y a vector in R$ , W1 = Span { u, uz}…

A: Given, {u1,u2,u3,u4,u5} be an orthogonal basis for R5,y a vector in R5,  W1=Span{u1,u2}   and…

Q: 7. If U¡¡ ‡ 0 are components of a tensor of the type (0, 2) and if the equation ƒU¡¡ + gU ji ³ = 0…

A: Given : Uij≠0    ,fUij+gUji=0 To Prove : f=g ,  Uij  is skew symmetric

Q: Answer the following questions about the Gram-Schmidt process for obtain- ing an orthogonal basis…

A: Since you have posted a question with multiple sub-parts, I will solve the first three subparts for…

Q: If x, x is a basis of solutions of r'y" + ary'+ by = 0, %3D then a = (A) -2 (B) 2 (C) -3 (D) 3 (E)…

A: Just use definition of solution of differential equations.

Q: 9. Let i, j denote the standard basis in R2. Find a scalar c if the vector 7i- 2j is perpendicular…

A: Dot product of the two vectors.

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: 2. 5. Consider twO basis of R: B = %3D -3 and (= 3. 5- 1- with respect to B, If a vector V E PR has…

A: The given 2 bases of ℝ2 are as follows. B=75, -3-1 and C=3-5, -12 Also, the coordinates of the…

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…