8. Let V be a vector space over F and v₁, ₂, € V. Then (₁₁) = (a₁v₁ + + a₂v, la,,a,F) is a subspace of V.

Q: Determine whether W is a subspace of the vector space V.W = {(x, y): y = ax2}, V = R2

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Q: - (-, s) Find a basis for the subspace W C R', consisting of all the vectors orthogonal to the…

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Q: If V is vector space, with u = (u, , U2) and v = (v, , v2) with u +v = (u,+v, , u2+v2) and ku =…

A: u=(u1,u2) , ku=(3u1,0) k=2, u=(-1,2)

Q: Q: Let W, ={(a, B.Y, 8): a=y+28, y= 28} and W2 {(a, B.Y, 8):a-y-0} be two subspaces of V,(R). Find a…

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Q: Let ū = (4, –2), ū = (-2,0), and w = (2, 2). Find the vector x that satisfies Tu – u +x = 5x + w. In…

A: Given Let Plugging the values in the equation:

Q: Let W be the subspace spanned by the vectors (1,2, 0, 3) and (2, 1,3,0). Find : a basis for W+.

A: Given that W is a subspace spanned by the vectors 1 , 2 , 0 , 3 and 2 , 1 , 3 , 0 . i.e., W = span1…

Q: Let W be the set of all vectors of the form a b where a - C = 4 and b - C = - 5. Lc Then, W is a…

A: Given: W=abc|ac=4, bc=-5 To prove W is subspace or not.

Q: 18 as a linear combination of a and y Express the vector v y.

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Q: Find the dimension of the subspace spanned by the given vectors: u₁ = (1, 3, -2,5), u₂ = (0, 1,-1,…

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Q: 5. Let W be the subspace of the vector space V = R spanned by the vectors =(1,3, 1, -2), B = (-3,…

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Q: Let W be the subspace spanned by the u's. Write y as the sum of a vector in W and a vector…

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Q: Let U and V be the subset of R' defined as U = { (a,b.c,d) : 3b-2d-a = 0 } and / = { (a,b,c,d) : a=c…

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Q: Q1. (a). Let V = R and define addition and scalar multiplication by: (a, b) e (c, d) = (a +2c,…

A: See the detailed solution below.

Q: 6. Let V be a vector space and v1, v2, V3, V4 are vectors in V. Suppose that {v1, v2, V3} are linear…

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Q: (a) Show that the vectors v1 = (1, 2, 3, 4), v2 = (0, 1, 0, – 1), and v3 = (1, 3, 3, 3), form a…

A: (a) The given vectors are: v1=(1,2,3,4), v2=(0,1,0,-1), v3=(1,3,3,3)

Q: Use the function to find the image of v and the preimage of w. T(V1, V2, V3) = (4v2 – V1, 4v, +…

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Q: Let W be the subspace spanned by u₁ and u2, and write y as the sum of a vector in W and a vector…

A: Let W be the Subspace Spanned by u1 and u2,Write y as the sum of a vector in W and a vector…

Q: Consider the vector space V defined by V = {(x, y, z) | x+y+z=0}. Let u = (1, -2, 1) and v = (3, -1,…

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Q: 2. Let S = span{V₁, V2, V3}. Check whether the vector u € St. (a) v₁ = [1, 0, 0, 0], v2 = [0, 3, 0,…

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Q: Let B' = {u1,u2} and ß = {v1,v2} be ordered bases for a 2-dimensional vector space V. Given that [u1…

A: Given u1+u2β=5,4t, and the change of coordinate matrix Q from β' to β is Q=1c3d. Express v1 as a…

Q: 9. Determine whether the vectors v₁ = (1,1,2), v₂ = (1,0,1) and v3 = (2,1,3) span the vector space…

A: We know that the dimension of vector space  is 3.If the given three vectors are linearly…

Q: 2. Consider the vectors [2, 1,0, 0], [3, 1, 0, 0], [0, 0,0, 1)] in R'. Find a vector v so that v,…

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Q: 13.) Determine whether the vectors span the vector space. (a.) Let u = 25t+6t²-t³,v=3+2t-4t2 (b.)…

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Q: 49. Show that in the vector space V, (R), the vectors (1, 2, 3), (- 2, 1, 4), (- 1, – 1/2, 0) form a…

A: We prove that the set of vectors form a linearly dependent set.

Q: If V is a vector space then show that cV=V.

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Q: 1) LetF1 be the subspace generated by the vectors (0, 1, 1), (0, 0.1) and 2 the subspace generated…

A: In this question, concept of subspace is used. Subspace A subspace is a vector space that is…