Which of the following are subspaces of P_2 with usual operations.pdf
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The document discusses the criteria for various sets to qualify as subspaces of the polynomial space p_2. It analyzes three sets: one is proven not to be a subspace, while another set formed by the span of {1+x, 1+x^2} is confirmed to be a subspace. The document concludes that the set formed by {1, x, x^2, x^3} does not constitute a subspace as p_2 is not contained within it.
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Which of the following are subspaces of P_2 with usual operations.pdf
- 1. Which of the following are subspaces of P_2 with usual operations? {1 +ax: x belongs to R} span {1 +x, 1+ x^2} span {1, x, x^2, x^3} = P_3 Solution 1. Consider two elements: 1+ax,1+ay with x,y in R For this set to be subspace sum of theset two elements must be in the set 1+ax+1+ay=2+a(x+y) which is not in the set Hence not a subspace 2. Let, P,Q be in this set so : a,b,r,s are scalars so that P=a(1+x)+b(1+x^2) Q=r(1+x)+s(1+x^2) P+Q=(a+r)(1+x)+(b+s)(1+x^2) which is in the set Let, P be in the set as before and c be a real number cP=ca(1+r)+cb(1+x^2) Hence, cP is in the set Hence it is a subspace 3. No it is not a subpace as P2 is properly contained in this set