Let W= span{(1,2,-1,0),(2,2,0,1),(1,1,-1,0)}. Use Gram Schmidt to find an orthogonal basis for W.

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: The vectors u1 = (1, 1, 1, 1), u2 = (0, 1, 1, 1), u3 = (0, 0, 1, 1), and u4 = (0, 0, 0, 1) form a…

A: Given,u1 = (1, 1, 1, 1), u2 = (0, 1, 1, 1), u3 = (0, 0, 1, 1), and u4 = (0, 0, 0, 1)Consider the…

Q: Let W be a space with an orthogonal basis given by and the shortest distance from W to y. Note: To…

A: Given that w=1-10011,111100.001-1-11y=-44431-4closest vector y to W is…

Q: Find a standard basis vector that can be added to the set {v1, v2} to produce a basis for R° a. V1 =…

A: To find standard basis vector

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…

Q: Consider the line L spanned by {(1, 2, 3)}in R. Find a basis of L-. a) {(0, –3, –2)} b) {(1,0,0),…

A: L is a line spanned by a single vectors  in R³, so perpendicular of L has dimension 2.

Q: Write down a basis for the vector space L(R", R).

A: Given vector space is the finite dimensional vector space of dimension m.

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: Let {u1, u2, u3} be an orthonormal basis for an inner product space V. If v = au1 + bu2 + cu3 is so…

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Q: The set S {uj = (2,0), u2 = (1,3)} is a (1, 3)} is a %3D %3D basis of R2. By applying Gram-Schmidt…

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Q: Let {u1, u2, u3} be an orthonormal basis for an innerproduct space V. If x = c1u1 + c2u2 + c3u3 isa…

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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: Show that the vectors a, (1, 0, 0), a2 = (1, 1,0), a, = (1, 1, 1) form a basis of C'(C).

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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 the orthogonal complement W⊥ of W and give a basis for W ⊥

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Q: Let (e1, ... , en) be a basis for a vector space V. Show that (e1 + en, e2 + en, en-1+ en, en) is…

A: The solution is given as

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: Let u2 u3 be a basis for R. Use the Gram-Schmidt process to find an orthogonal basis under the…

A: Here to find value of constants first we have to find out orthogonal basis using Gram-Schmidt…

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 the vectorsV1,..., Vk form an orthonormal basis of OC", then they also form a basis of R*. O True…

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Q: Whether vectors are u=(1, 2, 1), v= (4, 0, -4) and w=(1, -1, 1); determine if they are an orthogonal…

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Q: Find the coordinates of (2, 3, 4, – 1) with respect to the basis B = {(1, 1, 1, 2), (1, –1, 0, 0),…

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Q: Let e1, e2, e3, e4, ē5, e6} be the standard basis in R°. Find the length of the vector * = 471+…

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Q: Let {ui u2 be a basis for R?. Use the Gram-Schmidt process to find an orthogonal basis under the…

A: The given vectors u1=-30 and u2=-2-3

Q: Find an prthonormal basis of the plane x1+ 2x2- x3 0.. W

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let S = {u1, uz2, U3, U4} C Rª, where (1,1, 1, 1), U2 = (1, 1, – 1, –1), (1,—1,1, —1), и4 — (1, –1,…

A: As per the guidelines I solved your first three subparts of the question. For more solutions please…

Q: Let W= spana,3,0,-17,<1,3,0,-17, 23,6,0,-273, Find a basis for w. Furicd a basis for Wt. Verify the…

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Q: Let A = -1 -9 3

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

A: Use the below stated formula to find the required.

Q: Show that S = {(i, 0, 0), (i, i, 0), (0,0, i)} is a basis for C

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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: If {f1, . . . , fn} is an orthogonal basis of R^n and U = span{f1, . . . , fm}, show that U^⊥ =…

A: We will be proving the statement using basic definitions of Orthogonal Basis of Rn .

Q: Let B be a basis for a vector space V. Then vectors u, v, w in V are linearly independent if and…

A: Let V be a vector space of dimension n with basis B=b1, ⋯, bn. Let u=u1b1+⋯+unbn, then the…

Q: Let1, u2, U3, U4, U5} be an orthogonal basis for R5, y a vector in R5, W = Spantu1, uz} and W2 =…

A: Given a orthogonal basis

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: Let {u1, u2 U3, U4, U5} be an orthogonal basis for R5, y a vector in R5, W1 = Span{u1, uz} and W2 =…

A: For orthonormal basis of R5, {u1, u2, u3, u4, u5}  We know that span of the elements is orthogonal.…

Q: Let W = span{(1,1,-1), (2,3,1), (5,6, -2)}. i Find a basis for W. ii. Determine whether the vector X…

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Q: Let V1 = [0, 0, 5, 0], V, = [3, 4, 0, 0] and V3 = [4, 0, – 2, – 4] Use the Gram-Schmidt procedure to…

A: According to our guidelines we can answer only one question and rest can be reposted.

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

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Q: The Gran-Schmidt process t ransforns the basis u (1,2). uz-(2,-6) of R2 into the orthogonal basis V1…

A: The basis u1=(1,2), u2=(2,-6) of R2 into v1=u1=(1,2),we have find v2

Q: Use the Gram-Schmidt process to transform the given basis {w1, W2, W3} into an orthogonal basis…

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Q: Find a standard basis vector that can be added to the set {V1, v½} to produce a basis for R а. 0 —…

A: We have to solve given problem:

Q: Show that B = {(1,1,1), (1, –1,0), (1,1, –2)} is an orthogonal basis of R³ and then expand x = (a,…

A: Let A be a matrix with the given vector as row.

Q: 2. Consider the set (a) Verify that this is an orthogonal basis of R³. (b) Write the vector v = as a…

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Q: Let -3 be an orthogonal basis for a

A: Introduction: It is given that 4-30, 005 is an orthogonal basis for a subspace W. We have to find an…

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

A: Let  V be a vector space with an inner product . Suppose u1 , u2,..., un is a basis for V. Let…

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: Let u and v be linearly independent vectors in Rn that are not orthogonal. Describe how to find the…

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Q: Find a subset of u1, uz, uz, U̟ that gives a basis for W = span(u;) of R°, where %3D (а)) иј 3D %3D…

A: Given that u1=1,1,1,2,3 u2=1,2,-1,-2,1u3=3,5,-1,-2,5u4=1,2,1,-1,4 find a subset