Basis Problems
Example #1: Polynomials and determining bases for them
- Part 1:Is the set of polynomials $ x^2, x, 1 $ a basis for the set of all polynomials of degree two or less?
- quick solution
- rigorous solution
- Part 2: is the set of polynomials $ 3x^2, x ,1 $ a basis for the set of all polynomials of degree two or less?
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- Part 3: is the set of polynomials $ 3x^2 + x, x , 1 $ a basis for the set of all polynomials of degree two or less?
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- Part 4: is the set of polynomials $ 3x^2+x+1, 2x+1, , 2 $ a basis for the set of all polynomials of degree two or less?
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- Part 5: is the set of polynomials $ x^3, x, , 1 $ a basis for the set of all polynomials of degree two or less?
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- Part 6: is the set of polynomials $ x^2,3x^2, x , 1 $ a basis for the set of all polynomials of degree two or less?
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- Part 7: is the set of polynomials $ x^2,3x^2 + 1, x , 1 $ a basis for the set of all polynomials of degree two or less?
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- Part 8: is the set of polynomials $ x^2, x , 1 $ a basis for the set of all polynomials of degree THREE or less?
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- Part 9: is the set of polynomials $ x^2, x , 1 $ a basis for the set of all polynomials of degree ONE or less?
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- Part 10: is the set of polynomials $ x^2, x , 1 $ a basis for the set of all polynomials of EXACTLY degree TWO?
- solution: NO, it is not a basis for the set of all polynomials of exactly degree two. $ 0*x^2+0*x+0*1=0 $ 0 is not a polynomial of degree two. So this set of polynomials spans outside of the given space of polynomials.
- Part 11: is it possible to make a basis for the set of all polynomials of EXACTLY degree TWO?
- solution: Nope. Every basis can make the 0 vector
- Part 12: is the set of polynomials $ x^2, x , 1 $ a basis for the set of ALL POLYNOMIALS?
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- Part 13: Is it possible to make a basis for the set of ALL POLYNOMIALS?
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Example #2: Matrices
