11. Fix some positive integer n, and some a E R. The set {ƒ ≤ P„(R) | ƒ(a) = 0} W = is a subspace of Pn (R), the space of polynomials with real coefficients of degree ≤n. (You do not need to prove this.) Determine the dimension of W.

20. Let f be a continuous linear functional defined on the normed linear space X. Show that M = {xe X\f(x) = 0) is a closed subspace of X.

Solve all with handwriting solution else downvote definitely.

For each of the sets below, determine if the set is a vector space or subspace. a. H = {2], a, b real} а. b. V = , а — b {E 1.a, b real} С. T= 0 b. C = {the set of all complex numbers of the form a + bi, where a, b are real} e. J= {the set of all polynomials such that p(t)divides evenly by (t – 1)} [Hint: write p(t) in factored form, with the factor (t – 1) pulled out. What does the other factor look like?] d. - {the set of all odd functions: f (-x) = -f(x)} W is the set of all nxn matrices such that A² = A. h. Q is the set of all exponential functions i. f. g. S is the set of all n x n singular matrices

Select ALL sets that are not subspaces of R². The set containing the points on the graph of the line y = 2x – 13. The set R². The set {[_²2,3]} 13 The set containing all multiples of [ 2.83] 13 The set The set containing the points on the graph of the line y = {[:]}. 2 13 x.

Determine if the given set is a subspace of the appropropriate R^n and justify your conclusion.

Please show work

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