Thread: Why these polynomials p_1, p_2, p_3, p_4 doesn't span space of degree 2 polynomial?

We know the definition of span:
If is a set of vectors in a vector space , then the subspace of consisting of all linear combinations of the vectors in is called the space spanned by , and we say that the vectors span . To indicate that is the space spanned by the vectors in the set , we write:

or


I've a math problem and asked to find out whether the polynomials span a space or not. Determine whether the following polynomials span (the set of degree 2 polynomials).



The book says it doesn't span


How do they figure out that subspace does not consist of all linear combination of polynomials ? How does one come to this conclusion?

Originally Posted by x3bnm

We know the definition of span:
If is a set of vectors in a vector space , then the subspace of consisting of all linear combinations of the vectors in is called the space spanned by , and we say that the vectors span . To indicate that is the space spanned by the vectors in the set , we write:

or


I've a math problem and asked to find out whether the polynomials span a space or not. Determine whether the following polynomials span (the set of degree 2 polynomials).



The book says it doesn't span


How do they figure out that subspace does not consist of all linear combination of polynomials ? How does one come to this conclusion?

You must mean whether they span or not the space of all polynomials of degree 2 or less.

Just write each given polynomial as a vector , say , etc.

Now form a 3 x 4 (or 4 x 3) matrix with the above vectors and check you get a rank 2 matrix, thus 2 vectors out of

the four are linearly independent and thus they span a 2-dimensional space, which thus

cannot be all of P_2 since this space'sdimension is 3 .

Tonio

Never mind. I've found the answer.



Set up a matrix and solve for and that's it. If you have a valid solution the vectors span and if not vectors doesn't span.

Sorry tonio. May be we've posted at the same time. Thanks for help and explanation.