| Line 1: | Line 1: | ||
=Basis Problems= | =Basis Problems= | ||
| − | Example #1: Polynomials | + | Example #1: Polynomials and determining bases for them |
| − | :Part 1:Is the set of polynomials <math> x^2, x, | + | :Part 1:Is the set of polynomials <math> x^2, x, 1 </math> a basis for the set of all polynomials of degree two or less? |
:::quick solution | :::quick solution | ||
| Line 10: | Line 10: | ||
| − | :Part 2: is the set of polynomials <math> 3x^2, x | + | :Part 2: is the set of polynomials <math> 3x^2, x ,1 </math> a basis for the set of all polynomials of degree two or less? |
:::quick solution: | :::quick solution: | ||
| Line 16: | Line 16: | ||
:::rigorous solution: | :::rigorous solution: | ||
| − | :Part 3: is the set of polynomials <math> 3x^2 + x, x | + | :Part 3: is the set of polynomials <math> 3x^2 + x, x , 1 </math> a basis for the set of all polynomials of degree two or less? |
:::quick solution: | :::quick solution: | ||
| Line 22: | Line 22: | ||
:::rigorous solution: | :::rigorous solution: | ||
| − | :Part 4: is the set of polynomials <math> 3x^2+x+1, 2x+1, | + | :Part 4: is the set of polynomials <math> 3x^2+x+1, 2x+1, , 2 </math> a basis for the set of all polynomials of degree two or less? |
:::quick solution: | :::quick solution: | ||
:::rigorous solution: | :::rigorous solution: | ||
| − | :Part 5: is the set of polynomials <math> x^3, x, | + | :Part 5: is the set of polynomials <math> x^3, x, , 1 </math> a basis for the set of all polynomials of degree two or less? |
:::quick solution: | :::quick solution: | ||
:::rigorous solution: | :::rigorous solution: | ||
| − | :Part 6: is the set of polynomials <math> x^2,3x^2, x | + | :Part 6: is the set of polynomials <math> x^2,3x^2, x , 1 </math> a basis for the set of all polynomials of degree two or less? |
:::quick solution: | :::quick solution: | ||
:::rigorous solution: | :::rigorous solution: | ||
| − | :Part 7: is the set of polynomials <math> x^2,3x^2 + 1, x | + | :Part 7: is the set of polynomials <math> x^2,3x^2 + 1, x , 1 </math> a basis for the set of all polynomials of degree two or less? |
:::quick solution: | :::quick solution: | ||
:::rigorous solution: | :::rigorous solution: | ||
| − | :Part 8: is the set of polynomials <math> x^2, x | + | :Part 8: is the set of polynomials <math> x^2, x , 1 </math> a basis for the set of all polynomials of degree THREE or less? |
:::quick solution: | :::quick solution: | ||
:::rigorous solution: | :::rigorous solution: | ||
| − | :Part 9: | + | :Part 9: is the set of polynomials <math> x^2, x , 1 </math> a basis for the set of all polynomials of degree ONE or less? |
| + | :::quick solution: | ||
| + | :::rigorous solution: | ||
| + | |||
| + | :Part 10: is the set of polynomials <math> x^2, x , 1 </math> 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. <math> 0*x^2+0*x+0*1=0</math> 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 <math> x^2, x , 1 </math> a basis for the set of ALL POLYNOMIALS? | ||
| + | :::quick solution: | ||
| + | :::rigorous solution: | ||
| + | |||
| + | :Part 13: Is it possible to make a basis for the set of ALL POLYNOMIALS? | ||
| + | :::solution: | ||
| + | |||
| + | |||
| + | Example #2: Matrices | ||
Revision as of 09:11, 12 March 2013
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?
- quick solution:
- rigorous solution:
- 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?
- quick solution:
- rigorous solution:
- 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?
- quick solution:
- rigorous solution:
- Part 5: is the set of polynomials $ x^3, x, , 1 $ a basis for the set of all polynomials of degree two or less?
- quick solution:
- rigorous solution:
- 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?
- quick solution:
- rigorous solution:
- 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?
- quick solution:
- rigorous solution:
- Part 8: is the set of polynomials $ x^2, x , 1 $ a basis for the set of all polynomials of degree THREE or less?
- quick solution:
- rigorous solution:
- Part 9: is the set of polynomials $ x^2, x , 1 $ a basis for the set of all polynomials of degree ONE or less?
- quick solution:
- rigorous solution:
- 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?
- quick solution:
- rigorous solution:
- Part 13: Is it possible to make a basis for the set of ALL POLYNOMIALS?
- solution:
Example #2: Matrices
