10. Prove that W₁ {(a₁, a2,..., an) € F": a₁ + a₂ + + an = 0} is a subspace of Fr, but W₂ = {(a1, a2,...,an) € F: a₁ + a₂ ++an = 1} is not. =

Q: 3. Show that the set W = {(x, 0) : x e R} is a subspace of R?.

A: In order to show W is a subspace of R2 , we need to prove W is closed under addition. For any two…

Q: rove that W2 = {(a,b) E R² : a = 26²} is not a subspace of R?.

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Q: 4. Show that the set S = {ax²+bx+c: a, b, c € R and a ‡ 0} is NOT a subspace of R[x] by showing that…

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Q: 2. Let а — lb +c а S = {[a – b ,b,c € R a a. Show that S is a subspace of M2x2. 31 -2 31 5 b. Is |…

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Q: 4.) Determine the dimension of thr subspace of M21 consisting of vectors of the form [a – 2b] a + b…

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Q: Let H = {[ 21 12 x1, x2 € R 1,2 Show whether or not H is a subspace of R3.

A: We satisfy the clouser and scaler multiplication and show that 0 vector is also in H so H is a…

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Q: Show that {f in C[a,b]:f(a)=f(b)} is a subspace of C[a,b].

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

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Q: (d) The set U = [Select] C : a, b, c, d e R, ab = cd is a subspace of R¹.

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Q: Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1,y1)…

A: Given  be the set of all pairs  of real numbers together with the following operations:

Q: 4. Decide which of the following sets of vectors are subspaces of R2. Justify your answers. a₁ such…

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Q: Prove that W1 = {(a,b) E R² : 4a – b= 0} is a subspace of R2.

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Q: Determine whether W is a subspace of V

A: Given

Q: If V is a linear subspace of R", and x1 , ×2 , ... , Xn E V, then we must have span {x1, X2, Xn} =V.…

A: To determine that the statement, if V is a linear subspace of ℝn and x1, x2,....,xn∈V, then spanx1,…

Q: Determine If The Subset W Is A Vector Subspace Of V = R3 w = {(x, y, z): x * y-z = 0

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Q: If W is a subspace of R" of dimension k with a basis {v1,e,2,..., v%} then W Range(|v1 v2 vn)). ...…

A: Given: Let W be a subspace of Rn and dimk=k Let B be a basis of k then B=k Also B is a linearly…

Q: Q2. Let V = {(a, b, c, d) : b + 2c + d = 0} and W = {(a,b, c, d) : a + 2b = 0, c = 5d} be two…

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Q: 15 Suppose U is a subspace of V. What is U + U?

A: Let V be a vector space and U be it's subspace. Then , for all u,v ∈ U  u+v∈U and  cu ∈ U ,where c…

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: 9. Let F be a field. Find bases of the following two subspaces of F5 : W₁ = {(a₁, a2, a3, a4, a5) =…

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Q: 2. Let V = E3 and W = {x|x = a(1,0,2) + b(1,-1,3)} and S = {x\x = (x1,x2, 1)} where a and b are any…

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Q: Let A = X = 1 −1 1 1 −1 1 1 -1 1 -1 −1 1 1 1 -1 - Find a non-zero vector x in the null space of A.

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Q: 1 = { x:= = < |x|<1} am an 1. Consider the set Y=[-1,1] as a subspace of R. Is A=< open set in Y? Is…

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Q: Prove that span (v1,v2,...,vn) is a valid subspace of R

A: We know that ,  spanv1 , v2 , .... , vn = a1v1 + a2v2 + ...... + anvn / a1 , a2 , ...... , an∈ℝ A…

Q: Consider the vectors v1, v2, ..., Um in R". Is span (01,..., īm) necessarily a subspace of R" ?…

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Q: 4. Let H be the set of 2 × 2 whose trace is equal to zero. That is [a11 a12 Н- E M2(R) | a11 + a22 =…

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Q: Which of the following subsets of R* are subspaces of R*? b. W = {(a,b,c,d)| c=a+2b, d=a-3b}

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Q: Let W be a subspace of R". A set of vectors {v1, ..., Úp} in W is defined to be a basis of W if {v1,…

A: Let ℝn be a vector space. Let W be a subspace of ℝn. Then,  is W={v1→,......,vp→}a basis. Definition…

Q: 1. Is the subset V {(x, y, z, w), y = 3z, w = 2x + 1) a subspace of R*? Show the details of %3D your…

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Q: Let S = {E} " +=0}. such that ₁ + X2 + X3 = Either prove that S is a subspace, or find a…

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Q: 3. Determine, with proof, whether each set is a subspace of the given vector space {a, + a1x + a2x²…

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Q: Let V be the vector space of polynomials in t of degree ≤ 3. Determine whether the following vectors…

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Q: Let M be the set of all vectors (or polynomials) x in p (over R) for which x(t)=x(-t) - even…

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Q: Q2. Let V be a vector space over a field (F, +, .), let M1 , M2 and M3 be subspaces of V. Prove or…

A: V be a vector space over a field (F,+, .) , M1 ,M2 and M3 be the subspace of V

Q: c) Determine whether W = {(a, b, c, d) E R*| ad – bc = 0} is a subspace of R*. %3D

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Q: 1- Show whether the set of polynomials a, +at +at+at ,where a, =a, +a, is a subspace of P3 or not ?

A: Let V be a vector space over the field F. A non-empty subset S of the vector space V is said to be a…