MTH 421 Midterm: mth309
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Throughout the exam we de ne the set of natural numbers by n = Show all your work and justify your answers. (1)[14pts] consider the set of polynomials. B = {x3 + 3x2, x2 + 3x, x + 3, x3 + 4x2 + 4x + 4} p3. Show that b is a basis of p3. (2) let v be a nite dimensional vector space and s, t v subspaces of v . Show: (a) [9pts] s t is a subspace of v . (b) [6pts] dim(s t ) dim(t ). (3) consider the set. | a, b, c, d r and a b + 2c + 3d = 0 . Show your list is linearly independent. (d) [3pts] conclude that s has dimension three. (4) let v be a vector space and {v1, . , vn+1} v a set of linearly independent vectors of v . Show directly: (don"t just quote a theorem!) (a) [8pts] the set {v1, .