Prove that E = {(x, y, z) E R°, x + y+ z = 0} is a subspace of R3 and find a basis of E. Deduce dimE.

Q: e set B = {v= (1,1,1 a basis of R. 3. Linearly depende

A: Given set B = { v = ( 1, 1, 1 ), u = ( 0, 0, -3 ),  w = ( 2, 2, 0 ) } Then B is: -  A. a basis of R.…

Q: Find a basis for the subspace S of R4 consisting of all vectors of the form (a + b, a − b + 2c, b,…

A: We have to find the basis for the subspace S. We have to find the dimension of S.

Q: Given the basis {x,, X, cohere X,= 「に XgJ for R3 2 3. use the Gram - Schmidt process tO transform…

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Q: Let f E P(R) be a polynomial of degree n. Let yo,..., Yn be the dual basis of 1, x, x²,..., x". Show…

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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: Basis of vector space ; Suppose we have a set S , then it is basis of V if and only if : S is…

Q: Let S = {i, j} is a basis of R² and let W = {(x, y), x + y = 0} a- Show that W is subspace of R², b-…

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Q: In the vector space P, of polynomials of degree at most two, 2' transition of the matrix from the…

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Q: In the vector space p of polynomials of degree atmost two, the first column of the matrix of…

A: We have to choose the correct answer from the given options according to the given information

Q: Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: R2 →…

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Q: Let W = {a + bx + cx? + dx³ € Pg : c = a – b+ 2d} be a subspace of P3. Then a basis for W is:

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Q: Find a basis for subspace S of M2,2. s ={ [ ] : 4a – b = 2c + 2d ; a, b, c, d E R - C

A: We have to find basis of subspace S of M2x2 matrix.

Q: Find a basis of the subspace of R" that consists of all vectors perpendicular to both 1 and 1 -5 5…

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Q: and let W the subspace of R" spanned by i and v. Find a basis of W-, the orthogonal complement of W…

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Q: Find an orthogonal basis for the subspace {(x, y, z) E R³ | x + 4y – 2z = 0} -

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Q: Find an orthogonal basis of the subspace of spanned by the vectors (a) (b) -2 (c) (d) 234

A: To find the orthogonal basis of the subspace of spanned by the vectors 234 and 123 Consider that the…

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: Suppose the set ​{u​, v​} forms a basis for a subspace W in Rn.  Let x​=u​+v and y​=u​-v.  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: 2. Find the matrix of the reflection operator with respect to the plane x+y =0 (in the basis i, j.…

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Q: 3.8 Using the Gram-Schmidt method, turn the basis B = (b1, b2) of a two- dimensional subspace U C R…

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Q: Find a basis for the subspace and state the dimension. {[ 2c a − b b − 3c a + 2b ]:a, b, c in R}.

A: The given subspace is 2ca-bb-3ca+2b. 2ca-bb-3ca+2b=a0101+b0-112+c20-30 Note that, the vectors…

Q: Find a basis for the subspace a W = E R2x2 | 6a + 2b – c = d of R2x2- A basis for W is 1 0 0 Ex: 1 :

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Q: Find a basis for the subspace of R, the plane 2x +y-z+ 3w-0 and convert it into an orthonormal ban

A: Given the subspace 2x+y-z+3w=0. To get the orthonormal basis. The formula for orthonormal basis is,…

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: (a) Prove that the set ẞ = {p0(x), p1 (x), ... , Pn(x)} of Lagrange polynomials is linearly…

A: Given: ẞ = {p0(x), p1 (x), ... , Pn(x)} To Prove" (a). Prove that the set ẞ of Lagrange polynomials…

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

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Q: Given the inner-product = I p(x) q(x) dx for vector space P1 with basis set B={1, x}. Using…

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Q: Find an orthonormal basis for the subspace of C(R) generated by the functions {1, x, x²} with…

A: W e find an orthonormal basis for the subspace of C([0, 1]) generated by the function (1, x, x^2)

Q: Let w = ( { (x.y,z) ER3: x+y-z 0) be a subspace. A basis of w+ is [A].

A: Solution

Q: Let, T :R' → R';T(x,y,z) = (x – y,2x + 3y + 2z,x+ y+2z) 6. Determine the matrix representation of T…

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Q: Let, T:R'→R';T(x, y,z)=(2x-y+ z,x+2y-z,x+7y-42) Determine the matrix representation of T relative to…

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Q: If W is a subspace of ℝ4 whose dimension is 2 then is any set of 2 vector is a basis for W

A: Answer

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: Find a basis of a subspace U in R given by a system: I1 - T2 + 2r3+T1 T1- 12 + 313 + 2r, 0

A: (a) Given the system x1-x2+2x3+x4=0x1-x2+3x3+2x4=0 which describes the subspace U∈ℝ4.   To find the…

Q: Find a basis for the subspace W of R spanned by 3 7 2 2 10 6. 7 4 What is the dimension of W?

A: Given vectors, 122, 321, 11107, 764

Q: 2. Find a basis for the subspace W = {[r y z]T: ax + by + cz = 0, a, b, c e R} of R". What is the…

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Q: Suppose the set {u,v} forms a basis for a subspace w in Rn. Let x=u+v and y=u-v. Show that {x,y} is…

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Q: In the vector space P of polynomials of degree atmost two, the third column of the matrix of…

A: The solution is given below in the next step:

Q: Given the inner-product = I p(x) q(x) dx for vector space P1 with basis set B = {1, x}. Using…

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Q: Let W = {(a, b, c, d) E R*| a – b = 0,2b + c = d}. i. Show that W is a subspace of R*. ii. Find a…

A: We will prove the following result.

Q: Find a basis and the dimension of the subspace W = {(a+1)x+(2- )x²; a,b e R} of P,

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Q: In the vector space P, of polynomials of degree at most two, the third column of the matrix of…

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Q: Find a basis for the subspace of R W- (*,X,X3, X4, X )x + 2xz + Xg = 0) {(1.0.00.1).(.-1.0.0.2).…

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Q: In the vector space f, of polynomials of degree at most two, the third column of the matrix of…

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Q: Let U={p E p2 : p(2)=0}.Show that U is a subspace of p2 and find a basis for U.

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Q: A basis for the vector subspace: corresponds to? s = {(x, y, z) € R³ / x = 3y}

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Q: Find a basis and the dimension of the subspace W = {(a+1)x+(2-b)x²; a,b e R} of P,.

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Q: Find an orthogonal basis of the subspace of spanned by the vectors 目 目 [2] (a) (b) 3 (c) (d) 12 2 3…

A: We have to find an orthogonal basis of the subspace.