8. Prove that ū = (2,2, 1), ī = (-1,0,1) and w = (-2,3,0) are a basis for R³. Use these three vectors to construct an orthogonal basis for R³. %D

Q: 2) Show that S ER° defined by S = {(1,0, 5), (0, 2, -1)} is linearly independent. Extend S to be a…

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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: 1 Let C = [c1, c2, c3] = -1 2 3 -1

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Q: i) Find the matrix of T with respect to ordered basis B' = {(1, 1), (1, – 1);:.

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Q: 4. Find a basis for R3 that contains the vectors (1, 2, 3) and (3, 2, 1).

A: Note : according to our Company guidelines we can answer only first question, please repost the…

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 R^4 that includes these two vectors.

A: Solution: Given vectors are: v0=0342,  v1=1121 We know that the standard basis of ℝ4 is e1,e2,e3,e4…

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: 18. Use the Gram-Schmidt process to transform the given basis (w, w, orthogonal basis where w =…

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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: Q2. Leb u, and v be two livearly ondependent vecetors in R? Do thiy necussavily form a basis of R?

A: We have to write about vector u and v

Q: Let B = (1, x, x², x*) be an ordered basis for P3. Find the coordinate vector of 8x – 6x2 – 9x + 3…

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Q: Let B be the basis of R? consisting of the vectors 4 -2 and let C be the basis consisting of 3 2…

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Q: 4

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Q: Find a basis for the set of vectors in ℝ2 on the line y = 5x.

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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: Find a basis of R that contains the vectors X = (1,1,1,1) and Y = (1,1,0,1).

A: Given Vectors X =(1,1,1,1) and Y = (1,1,0,1) Now determine the orthogonal basis for R4

Q: 4. Present the standard basis vectors e, and ez in R? as linear combinations of (3, 1) and (1, 3).

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Q: 2

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Q: 5/ Find the coordinates of x = with respect to the basis consisting of uj = (1, 3, 2), u2 = (2,…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

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: 1. Show that the vectors (1, 1,0), (1, 1, 1) and (1,0, 0) are a basis of R3. Find the coefficients…

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Q: The Gram-Schimidt process transforms the basis u=(1, 3), u2-(1, -7) of R intothe orthogonal basis…

A: Given u1=⟨1,3⟩,u2=⟨1,−7⟩

Q: Show that {e1,€2, - } is a Schauder basis for P, 1<p< ∞, where S1 ifi=n (E") en = %3D 0ifi#n.

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Q: Let {e1, e2, ē3, ē4, ē5, ē6} be the standard basis in Rº. Find the length of the vector 4ē4 + 4ē, –…

A: Consider the given standard basis in R6 as, e1→,e2→,e3→,e4→,e5→,e6→ As we know that standard basis…

Q: now that B = {x, r³ + 2x² + 4, x² + 2, 1} is a basis for P3.

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Q: 13. In P2, find the change-of-coordinates matrix from the basis B = {1 – 21 + t², 3 – 5t + 41², 2t +…

A: To find - In ℙ2, find the change-of-coordinates matrix from the basis B = 1 - 2t + t2, 3 - 5t + 4t2,…

Q: 15. Use the Gram-Schmidt process to transform the basis {r1, x2, a3, T4} into an orthogonal basis…

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Q: The set S {u1 (2,0), u2 (1,3)} is a || basis of IR?. By applying Gram-Schmidt process to produce…

A: Given the set { u1, u2} is a basis of R2.

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: Determine whether the set S {1 – t, t, 1 + t²} a. Spans P2 b. Is linearly independent c. Is a basis…

A: As per our guidelines I am solving only 1st for others kindly resubmit the question mentioning that…

Q: 2 then the is the matrix of transition from the base B to the basis B' and the coordinates of vector…

A: Given that B and B' are bases of a vector space v and P is the matrix of transition from base B to…

Q: Find an orthogonal basis for R3 that contains the vector

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Q: 15. Use the Gram-Schmidt process to transform the basis {x1, 2, 23, x4} into an orthogonal basis for…

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Q: 2.) Determine whether S = is a basis for R³. If it is write u = (6,3,6) as a linear combination of…

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Q: Let B = {v1, v2, V3} be a basis for R3. For any u, v E R³ define || (u, v) = [u]E[v]B• (a) Is (, )…

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Q: The Gram-Schmidt process transformsthe basis u1=(-3,1), u2=(7, 1) of R´ into the orthogonal basis…

A: Orthogonal set problem....

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 the matrix for T relative to the basis B, and use Theorem 8.5.2 to compute the matrix for T…

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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: Show that B = {1,1 – x,2 – 4x + x²,6 – 18x + 9x² – x³} is a basis for P3(R). Hint: Recall that…

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Q: i) Find the matrix of T with respect to ordered basis B' = {(1, 1), (1, – 1)} %3|

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Q: Explain why S is not a basis for P2. S = {1 – x, 1 – x², -1 – 6x + 7x²} O sis linearly dependent.

A: Given that S=1-x,1-x2,-1-6x+7x2 Explain why S is not a basis for P2 if S is linearly dependent the S…

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: Write x as the sum of two vectors, one in Span {u, ,u2.u3} and one in Span fu4}. Assume that {u,…

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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: Explain why S is not a basis for P,. S = {1 – x, 1 – x2, -1 – 7x + 8x²} O sis linearly dependent. O…

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Q: Show that (u₁. U₂. u) is an orthogonal basis for R³. Then express x as a linear combination of the…

A: Answer is in next step. please give a like !!