1. In the following problems check to see if the set S is a subspace of the corresponding vector space. If it is not, explain why not. If it is, then find a basis and the dimension. (a) (b) (c) (d) X1 X2 S = X3 ᏆᎪ 2x1 + x2 x3; x1 + 2x2 + x3 = 0 CR X1 S = , x1 x2 CR² X2 a 0 Va ЄR CR2×2 { [8 , ], Va € R -a } S = R} C df 5= {f(x), // = sin(wf), Vw #0 € R 0ER CR S dx

Q: The vector x is in a subspace H with a basis B = {b₁,b2}. Find the B-coordinate vector of x. 1 -2 -6…

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Q: Which one of the following is a subspace of P3 (A) The set of all polynomials of degree 1 (B) {a +…

A: Given a vector space V, let W be subset of V , then W is called subspace of V iff If u and v are…

Q: Consider the set W of 2 x 3 matrices with the given constraints. a {[} ·] : ª = 0 b 0 c d W = : a =…

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Q: Let the vector v and the subspace W be as given below. 13 V = W = span -12 (a) Find a basis for W.…

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Q: (b) Define subset W in R³ as W = {[x1,x2, x3]|4x1 – x2 + 5x3 = 0}. W is a subspace of R3 -

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Q: For a basis for the following subspace of R4:

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Q: Either use an appropriate theorem to show that the given set, W, is a vector space, or find a…

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Q: Let W be the subspace spanned by the given vectors. Find a basis for W⊥

A: The orthogonal complement of the subspace W is the set W⊥=v∈V : v, w=0 for all w∈W and the…

Q: If S is a subspace of R", then so is the orthogonal complement of S.

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Q: (a) F = Q, V = R², and W = Z². (b) F = R, V = R², and W = {(x, y) E R? : ry > 0}. (c) F = R, V = R³,…

A: Let W≠0 be a subset of a vector space V over F with the same addition and scalar multiplication as…

Q: [a 2. Prove: W is set of all 3x2 matrices of the form: 0 Lo Is a subspace of M3,2 b a + 3b 2c

A: W is a set of all 3*2 matrices of the form  ab0a+3b0cis this a sub space of M3×2

Q: 3. Suppose that u and vare nonzero vectors and that S is a subspace. Prove that if u ES and v E St,…

A: Let u and v be non-zero vectors and S be a subspace. Assume that u∈S and v∈S⊥. This means that…

Q: Decide if the two subspaces U = span(U) and V = span(V) spanned by the two subsets and of vectors in…

A: (.)   Orthogonal subspace :  Two subspaces 'U' and  'V' are orthogonal to each other if  u , v= 0…

Q: The set of solutions of the system of linear equations x₁ + x2-x3-4 = 0 x1 + x2 + ix3 + ix4 = 0 is a…

A: We have to solve given problem:

Q: s) Determine if these vectors for a basis for R³: linear combination of these vectors. 0·0-0--0- and…

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Q: Computations: [7x -3y 9. Compute a basis of x, y E R x + y

A: We will find out the required basis.

Q: Let W be the subspace spanned by the given vectors. Find a basis for W-. 1 -1 1 W1 W2 -5 -2 1

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Q: The vector x is in a subspace H with a basis B = {b, ,b2}. Find the B-coordinate vector of x. %3D 1…

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Q: Exercise 4.3 Which of the given subsets of the vector space, M23 of all 2x3 matrices are subspaces?…

A: Vector subspace.

Q: Find the inverse of the following matrix by using Adjoint Method where |A| = 4c + 10. A = c+1 0 2 1…

A: If A is an invertible matrix, then its inverse will be:                      A-1=1AadjA where,…

Q: Find bases for the following subspaces of F5: and W₁ = {(a1, 02, 03, 04, 05) € F5: α₁ - 03 - α4 = 0}…

A: Given subspaces areWe have to find the bases for the above subspaces also we have to determine their…

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 of the following subspace of R³: W linear independence. {(a+b,b+c, c-a)}. Hint:…

A: Given Subspace is, W={(a+b,b+c,c-a):a,b,c∈ℝ}   Now, (a+b,b+c,c-a) =a(1,0,-1)+b(1,1,0)+c(0,1,1)   So,…

Q: 5.6. Are the polynomials 1-2x+3x², 2+x-x², 1-2x+x² linearly independent in the vector space P₂? If…

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Q: The vector x is in a subspace H with a basis ß = {b1,b2}. Find the ß-coordinate vector of x 9) b₁ =…

A: Introduction: To find the β-coordinates of the given vector corresponding to the the given basis…

Q: 2 In each of the following cases, determine whether the subset W is a subspace of the given vector…

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Q: Let D = (d₁d2₂ d3) and F = {f₁.f2.13) be bases for a vector space V, and suppose f₁ =3d₁-d₂ +5d3, f₂…

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Q: The vector x is in a subspace H with a basis B = {b₁,b₂}. Find the B-coordinate vector of x. 5 *H b₁…

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Q: 6. Show that the set of all 3 x 1 matrices is a vector space. **end**

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Q: Find a basis for the solution subspace Ax=0 where A=[-2,4; 1,-2] OA. {(1,0),(0,1)} O B. {(1,2)} C.…

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Q: how that if T is a linear operator on V and W1, W2 are T-invariant subspaces of V, en W1+ W2 is a…

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Q: Determine whether the vectors w₁ = [2,1,–3], w₁ = [4,0,2], and w3 = [2,-1,3] form a basis for the…

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Q: Consider the subspace W = {(a, b, c) E R³ : a – 26 – 3c = 0} of R³. (b) Is {(5, 1, 1)} a basis for…

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Q: The vector x is in a subspace H with a basis ß= (b₁, b2]. Find the B-coordinate vector of x. -6-0-3…

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Q: V2 4. Find a basis for the subspace V=- E R4: V3 v1 – 2v2 + 3v3 + V4 = 0

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Q: et B = and C = {c1,c2} be bases for a vector space V, and suppose b, = 8c, - 5c2 and b2 = - 7c, +…

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Q: 3. Consider V = {a2x² +a1x + ao € P2(R) : a2 + a1 + ao = 0} . Use the subspace theorem to show that…

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Q: Let W = {(x1, x2, x3) = R³ : x₁ + 2x2 − 3x3 = 0} x1 (a) Find a basis of W. (b) Give an example of a…

A: Answer follows

Q: 3. (a) Let W = {a+bx + cx² : ac= 0} C P2. Is W a subspace of P2? Justify your answer. (b) Let V = {a…

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Q: he vector x is in a subspace H with a basis B = {b₁,b₂}. Find the B-coordinate vecto b₁ X]B = 5 -7…

A: Lets say the B-coordinate vector of 'x' is