Problem 3. Let V, W be vector spaces over F, and let U be a subspace of V. Suppose SE L(U, W) is such that S 0. Define T: V → W by Prove that T is not linear. T(v) = JSv 10 if v EU, if v EV and v U.

Q: Find a vector b such that |b| = 1/2 that is parallel to a = but with opposite direction.

A: Given that: |b|=12a=<-6, 3, -2>

Q: Is Mmxn(Q) is a vector space over R?

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

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Q: Let x=⟨1,1,1⟩ and y=⟨0,3,−2⟩Find a unit vector that is orthogonal to both x and y

A: If w is an unit vector which is orthogonal to x and y, so

Q: defined as Let V = {(x,y) | x, y = R}, with vector addition and scalar multiplication (x₁, y₁) (x2,…

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Q: Let x 0 and y 0. (a) If xly, show that {x, y} is a linearly independ- ent set. (b) Extend the result…

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Q: 3. Let V and W be F-vector spaces, and let L(V, W) be the set of all linear transfor- mations from V…

A: Solution:

Q: Show that 0 x a =0 = a x 0 for any vector a in V3

A: lets take vectorv=(x,y,z) and   0=(0,0,0)

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: Let V be a vector space and let V1, V2, V3 be vectors in V. Define the set S and S' by S = {v1, V2,…

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Q: 10. Let the vector v = 2i -3j + k, and the vector w = = -i+3j+ 2k. Find v x w.

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

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Q: Q.5. Show that the vectors V1 = ( 1, 1, 2, 4 ), V2 = ( 2, -1, -5, 2 ), V3 = ( 1, -1, -4, 0 ) and V,…

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