For each pair of vector space (V, W) over R, either give an isomorphism V → W or show that no such isomorphism can exist

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For Each Pair Of Vector Space V W Over R Either Give An Isomorphism V W Or Show That No Such Isomorphism Can Exist 1 (86.1 KiB) Viewed 8 times

For each pair of vector space (V, W) over R, either give an isomorphism V → W or show that no such isomorphism can exist. (Here C[a, b] denotes the vector space of Cn functions [a, b] → R. Recall a function is of class C if it is n times differentiable and the nth derivative is continuous). (a) V = R¹, W = {x € R5 : x₁ + x2 +X3 + X4 +X5 = 0}. (b) V = R5, W = : P5(R). (c) V = Cº[0, 1], W = Cº[-1, 1]. (d) V = Cº[0, 1], W = {ƒ € C¹[0, 1]: f(0) = 0}. · (e) V = R², W (f) V = R¹, W = Cº[0, 1]. (g) V = R[x], W = RN. = {real-valued solutions to ï(t) + x(t) = 0}.