4. Let {V1,... , Vn} be an orthogonal basis of R". For any vectors x, y in R" show that (x · V;) (y · V;) 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: 2. Find the coordinate vector of & = (2, 3) relative to the basis B = {(1,0), (1, 1)} in vector…

A: As per your query the solution of the question (2) has been done. For the solution of the problem…

Q: 8. Given that B = {u, v, w} is a basis for a vector space V. Determine if the set 2u+v+w} is a basis…

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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 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: The Gram-Schmidt process transforms the basis u=(1, 2), U2 =(6, 2) of R into the orthogonal basis…

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

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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: 1. 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: 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: 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: The Gram-Schmidt process transforms the basis u,=(1, 3), u,=(1, –7) of R² into the orthogonal basis…

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Q: Define the basis B of R* by {0) 000 B = 1 What is the component vector [v]s of v = relative to the…

A: See solution below

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 S = {V1, V2} be a basis of R? where v1 and v2 = -3 What element of R? is determined by the…

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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: 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: Let W be the set of all vectors with z and y real. Find a basis of W [z +

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Q: Take L: V --> W, from P2 --> P2 and L(p) = p + p', with basis B = {1 + x + x2, 1 + x, 2x + x2} and…

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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: 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: 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: 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: The Gram-Schmidt process transforms the basis u3(1,-3), u,=(-7, 1) of R into the orthogonal basis…

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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: 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: The Gram-Schmidt process transforms the basis u1=(-1, 2), uz=(6, –2) of R into the orthogonal basis…

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Q: Let W be the plane 3x + 2y − z = 0 in R3. Find a basis for W⊥.

A: Let W be the plane 3x + 2y − z = 0 in ℝ3. Find a basis for W⊥. We know that dimension of W and…

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=(2,-6) of R into the orthogonal basis…

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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: 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: 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: 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: 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: 10)Define TE L(R²) by T(x, y) = (41x + 7y,-20x + 74y). Verify that the basis of T %3D with respect…

A: Question 10. The detailed solution is as follows below: