Problem 2 i) Show that u = {a(1+x)+b(1+x+x²): a, b = R} is a subspace of P2. ii) Recalling that every differentiable function is continuous, show that V = {f: [0, 1] R f is differentiable} is a subspace of C([0, 1]). Desired takeaways: Learning to apply the subspace criterion to show that a set is a subspace of a larger vector space.

Q: 1. Consider the following subspaces of R³: U = {(x,x,y): x,y ER} and W = {(x,-x, 2x): x, y ≤ R} (a)…

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Q: (1) Consider the space M2×2 (R) of 2 x 2 matrices equipped with the usual addition and scalar…

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Q: 4. Determine which of the following sets of n x n ma- trices are subspaces of Mnxn (R). a) The n x n…

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Q: The closest point is

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Q: 4. Determine whether W is a subspace of R³. a. W = {(x-1, x,x) x ER} b. W = {(3x, 0,-x)|x = R} 3 x 0…

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Q: True or False. Justify your answer. Let U Span{1+x, x+x²} and W = Span{x²} be subspaces of P2 (R).…

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Q: Q6: Explain whether the following sets are not subspaces of R" for the appropriate value of n. a a a…

A: We have to show that given set are vector subspace or not

Q: b) Is W = {f(x):f is a polynomial with degree <3,f (0) 0} a subspace of P: all 3 ? Explain your…

A: To see if the given subset is a subspace of the vector space of all polynomials of degree less than…

Q: 5. Let U = {(u₁, U2, U3) E R³|u₁ + u2 - u3 = 0} and V = {(V1, V2, V3) E R³v₁ - 3v2 +5v3 = 0}. (a)…

A: (a)

Q: 5. Let H = {x E R³| x1 + x2 + X3 = 0}. Prove that H is a subspace of R3.

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Q: The set of all vectors in R squared  that satisfy 3x+4y=1 is a subspace of R squared . True False

A: 1. The set of all vectors in R squared that satisfy  3x+4y=1  is not a subspace of R squared.  i.e.…

Q: True or False, S is a subspace of R 3 x1 (8] x2 x3 S = O True O False 2x1 + x3 = 0, x1 + x2 = x3 = 0…

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Q: Give a counter example to show that W is not subspace of R³. W=set of (x, y, z) where x + y = 1 6. ,

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Q: Show that P2 is a subspace of P3 (See polynomial vector spaces.) the definition of the

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Q: Determine whether the subset of C(-0, co) is a subspace of C(-0, co) with the standard operations.…

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Q: Find the closest point to y 2 4 0 in the subspace W spanned by ₁ = 2 0 -1 -3 and ₂ = 7 -2 3

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Q: 2

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