$U={ pin P_{4}left( mathbb{R} right) : p''left( 6right) =0}$ Find a basis for $U$
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Find a basis for $U$ , where $$U={ pin P_{4}left( mathbb{R} right) : p''left( 6right) =0}.$$ This question is 2.C 5 of Linear Algebra Done right. I would like to know how this person got the example basis: $${1, x, x^{3}-18x^{2},x^{4}-12x^{3}}.$$ My attempt: Take $ fleft( xright) = ax^{4}+bx^{3}+cx^{2}+dx + e$ $f''= 12ax^{2}+6bx+2c $ $implies f''left( 6right) = 0 implies c = -6^{3}a-18b$ $implies fleft( xright) = a(x^{4}-6^{3}x^{2})+b(x^{3}-18x^{2})+dx + e$ $implies$ my basis is ${1, x, x^{3}-18x^{2},x^{4}-6^{3}x^{2}}$ Where have I gone wrong?
linear-algebra proof-verification polynomials vector-spaces hamel-basis
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