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" Let P4 denote the vector space of all polynomials with degree less than or equal to 4 and real coefficients. Describe percisely as you can the linear span of set {x^2 – x^4, 1 + x^2, 2x^4 – x^2 – 5} "

To break it down in simple english, my prof said that you must subtract each part from each other until you get the span as simple as possible.

Example.. ( x^2 – x^4 ) - ( 1 + x^2 ) = ( -1 + -x^4 )

Now I don't understand WHY do I do that, and how it still keeps it as a Spanning Set? Could somebody please explain that.

So anyway, I did what the prof said, and simplified the spanning set.

I got.... P4 = span{ 1 + x^4, 1 + x^2, 6}

Now the second part that I don't understand is, HOW do i explain what this linear span means?

Can somebody please help.

Thanks