Which of the following are vector spaces over r prove each


Which of the following are vector spaces over R? Prove each answer. It may be helpfulin some cases to use what you know about subspaces

.(a) The integers Z = {...,-2,-1,0,1,2,...} with the usual operations of addition and multiplication

(b) The set R[x] = {anxn + an-1xn-2 + ··· + a1x + a0 | ai ∈ R} of all polynomials in the variable xwith real coefficients with the usual operations of addition and scalar multiplication, i.e.,for m ≤ n,(anxn + ··· + a1x + a0) + (bmxm + ···b1x + b0):= anxn + ··· + am+1xm+1 + (am + bm)xm + ··· + (a1 + b1)x + (a0 + b0)c ·(anxn + ··· + a1x + a0) := canxn + ··· + ca1x + ca0

(c) The set Tn(R) of all n × n diagonal matrices with matrix addition and scalar multiplication

(d) The set Bn(R) of all upper triangular n × n matrices with matrix addition and scalar multiplication

(e) The set GLn(R) of all n×n invertible matrices with matrix addition and scalar multiplication

 

 

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