Let be a subspace of an inner product space and let be the corresponding projection map. Which of the following are true? a. and U V TU : V → U ker(Ã) = U¹ ----

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Let V be a vector space with a scalar product. Show that = 0 for all v E V.

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

2) injective Let V be a real vector space. Prove that R linear map described below is V → Vc v→ 10v (This lets us "think" of V as being a subspace of Vc)