the 3d vector space of (at most) quadratic expressions in a variableThe space of (at most) quadratic functions in a single variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn is a 3-dimensional vector space: the sum of two quadratics is a quadratic whose ordered coefficients are the sum of the individual quadratics, while the ordered coefficients of a scalar multiple of a quadratic are just the scalar times the ordered coefficients of the quadratic.
It is natural to order the powers of the variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn in increasing order from 0 to 2 (so that we can easily extend this discussion to any number of powers by adding additional terms, as in Taylor polynomials). Technically a quadratic function must have the coefficient of its squared term be nonzero, otherwise it is a linear function or even a constant function, so we are talking about the space of "at most" quadratic polynomials, namely polynomials of degree at most 2. This will be understood here when we refer to these functions as quadratics.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To each quadratic corresponds a vector of ordered coefficients, and adding quadratics or scalar multiplying them corresponds directly to the same vector operations on the corresponding vectors: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The ordered set of functions 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 is a basis of this vector space since every quadratic can be expressed uniquely as a linear combination of these three functions, with respective coefficients traditonally called 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 These coefficients correspond respectively to the Cartesian coordinates 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 on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JVEiM0YnRjgvRjxRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRjhGQg==, while the basis 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 has coefficient vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNictRiM2Ki1GMDYnLUYjNiktSSNtbkdGJDYlUSIxRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGPEY/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjk2JVEiMEYnRjxGP0ZCRmVuRjxGP0Y8Rj8vJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnLUZDNi5RIn5GJ0Y8Rj9GRi9GSUY+RktGTUZPRlFGU0ZVL0ZZRldGQi1GMDYnLUYjNilGZW5GQkY4RkJGZW5GPEY/RjxGP0ZobkZbb0ZCLUYwNictRiM2KUZlbkZCRmVuRkJGOEY8Rj9GPEY/RmhuRltvRjxGP0Y8Rj8vRmluUSJ8ZnJGJy9GXG9RInxockYnRitGPEY/ and so corresponds to the usual basis { LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSZtb3ZlckdGJDYlLUYjNigtRiw2J1EiaUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGOS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRidGOi8lK2ZvcmVncm91bmRHUStbMCwxNjAsODBdRicvJSxwbGFjZWhvbGRlckdGOS8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGOS9GPVEnaXRhbGljRictSSNtb0dGJDYuUS0mUmlnaHRBcnJvdztGJ0ZFL0Y9USdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGUy8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0ZTLyUobGFyZ2VvcEdGUy8lLm1vdmFibGVsaW1pdHNHRlMvJSdhY2NlbnRHRlMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zcby9GaW5GOUZP, 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, 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} of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiM2JS1GLDYmUSgmcmVhbHM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGOS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGOkY8LUkjbW5HRiQ2JVEiM0YnRjpGPC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y6Rjw=. Thus we can picture each quadratic function as a vector (arrow) in space. The three functions 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 are linearly independent functions since if any linear combination of them equals the zero function 0, then all the coefficients have to be zero, and they certainly span the space of all quadratics by definition. Thus they satisfy the two properties of a basis.changing the basis using a Taylor polynomialHowever, we could also express any quadratic function in terms of its Taylor polynomial about a value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn different from 0, say LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8. This gives us another set of coefficients which are related to the old coefficients by a linear transformation, equivalently a new basis of this vector space with new coordinates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNictRiM2KS1GLDYlUSJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJCRidGN0Y6Rj0tRiw2JVEiQUYnRjdGOi8lK2V4ZWN1dGFibGVHRkVGQUZobkZBLyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJy1GPjYuUSI6RidGaG5GQUZDL0ZHRkVGSEZKRkxGTkZQL0ZTUSwwLjI3Nzc3NzhlbUYnL0ZWRmVvRmhuRkE=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY9LUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJyZwbHVzO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GUy1GLDYlUSJiRidGL0YyLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGUkZULUYsNiVRInhGJ0YvRjJGWC1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZYLUYsNiVRImFGJ0YvRjJGWC1JJW1zdXBHRiQ2JUZlbi1JI21uR0YkNiRRIjJGJ0Y5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlxwLUYsNiVRIkNGJ0YvRjJGNS1GLDYlUSJCRidGL0YyRlgtSShtZmVuY2VkR0YkNiQtRiM2J0Zlbi1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GYm82JFEiMUYnRjkvJStleGVjdXRhYmxlR0Y9RjlGOUY1LUYsNiVRIkFGJ0YvRjJGWC1GX282JS1GZXA2JC1GIzYnRmVuLUY2Ni1GW3FGOUY7L0Y/RjFGQEZCRkRGRkZIRlIvRk5RLDAuMzMzMzMzM2VtRidGXHFGX3FGOUY5RmFvRmVvLUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGW3BGXXAtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGUy8lJmRlcHRoR0Znci8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYjUSFGJ0ZfcUY5or multiplying out: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 can describe this as a change from the old basis functions 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 to using instead the new basis functions 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
The columns of the basis changing matrix are the old coordinates of the new basis vectors, which we can read off from their expanded form
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
hence the basis changing matrix with these 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 the old coordinates are the basis changing matrix times the new coordinates: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This relationship between the new and old coordinates is exactly what we get just by expanding the right hand side (by multiplying it out) of the equation between the two quadratic expressions and comparing coefficients on both sides of the equation, namely: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so identifying the coefficients of the powers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn on each side 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or finallyLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkobWZlbmNlZEdGJDYmLUYjNigtSSNtaUdGJDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYsNi1RIixGJ0YvRjIvRjZGUUY3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJy1GTDYlUSJiRidGT0ZSRlQtRkw2JVEiYUYnRk9GUkYvRi8vJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnLUYsNi1RIj1GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGZG8tRiM2Jy1GTDYjUSFGJy1JKG1hY3Rpb25HRiQ2JC1GRzYoLUYjNidGaG8tSSdtdGFibGVHRiQ2OC1JJG10ckdGJDYoLUkkbXRkR0YkNigtRiM2Ji1JI21uR0YkNiRRIjFGJ0YvLyUrZXhlY3V0YWJsZUdGNC8lLHBsYWNlaG9sZGVyR0ZRRi8vJSlyb3dhbGlnbkdGam8vJSxjb2x1bW5hbGlnbkdGam8vJStncm91cGFsaWduR0Zqby8lKHJvd3NwYW5HRmBxLyUrY29sdW1uc3BhbkdGYHEtRmlwNigtRiM2Jy1GLDYtUSomdW1pbnVzMDtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGZ3JGXXFGYXFGY3FGL0ZlcUZncUZpcUZbckZdckZocEZlcUZncUZpcS1GZnA2KC1GaXA2KC1GIzYmLUZecTYkUSIwRidGL0ZhcUZjcUYvRmVxRmdxRmlxRltyRl1yRmhwLUZpcDYoLUYjNidGY3ItRl5xNiRRIjJGJ0YvRmFxRmNxRi9GZXFGZ3FGaXFGW3JGXXJGZXFGZ3FGaXEtRmZwNihGW3NGW3NGaHBGZXFGZ3FGaXFGYXEvJSZhbGlnbkdRJWF4aXNGJy9GZnFRKWJhc2VsaW5lRicvRmhxUSdjZW50ZXJGJy9GanFRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0ZRLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGaHQvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGY3UvJSZmcmFtZUdGY3UvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0Y0LyUtZXF1YWxjb2x1bW5zR0Y0LyUtZGlzcGxheXN0eWxlR0Y0LyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRmB1RmhvRmFxRi9GLy9JK21zZW1hbnRpY3NHRiRRJ01hdHJpeEYnL0Zbb1EiW0YnL0Zeb1EiXUYnRmZ2LyUrYWN0aW9udHlwZUdRLnJ0YWJsZWFkZHJlc3NGJ0Zob0ZhcUYvRistRlxwNiQtRkc2KC1GIzYnRmhvLUZjcDY4LUZmcDYmLUZpcDYoLUYjNiYtRkw2JVEiQ0YnRk9GUkZhcUZjcUYvRmVxRmdxRmlxRltyRl1yRmVxRmdxRmlxLUZmcDYmLUZpcDYoLUYjNiYtRkw2JVEiQkYnRk9GUkZhcUZjcUYvRmVxRmdxRmlxRltyRl1yRmVxRmdxRmlxLUZmcDYmLUZpcDYoLUYjNiYtRkw2JVEiQUYnRk9GUkZhcUZjcUYvRmVxRmdxRmlxRltyRl1yRmVxRmdxRmlxRmFxRlt0Rl50RmB0RmJ0RmR0RmZ0Rml0Rlt1Rl51RmF1RmR1RmZ1Rmh1Rlt2Rl12Rl92RmF2RmR2RmhvRmFxRi9GLy9GZ3ZRKkNvbFZlY3RvckYnRml2Rlt3RmN5Rl13RmhvRmFxRi8=The columns of this basis changing matrix are the old coordinates of the new basis vectors exactly as we saw above.This is easy to invert by hand since the last equation is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5, then back subbing into the next to last equation 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and finally backsubbing both of these solutions into the first equation gives
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with inverse coefficient matrix orLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkobWFjdGlvbkdGJDYkLUkobWZlbmNlZEdGJDYoLUYjNigtSSdtdGFibGVHRiQ2OC1JJG10ckdGJDYnLUkkbXRkR0YkNiktRiM2JS1JI21pR0YkNiVRIkNGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGZ25Gam4vJStleGVjdXRhYmxlR0Y0LyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0Zgby8lK2dyb3VwYWxpZ25HRmBvLyUocm93c3BhbkdRIjFGJy8lK2NvbHVtbnNwYW5HRmdvRlxvRl5vRmFvRmNvLUZSNictRlU2KS1GIzYlLUZaNiVRIkJGJ0ZnbkZqbkZnbkZqbkZcb0Zeb0Zhb0Zjb0Zlb0Zob0Zcb0Zeb0Zhb0Zjby1GUjYnLUZVNiktRiM2JS1GWjYlUSJBRidGZ25Gam5GZ25Gam5GXG9GXm9GYW9GY29GZW9GaG9GXG9GXm9GYW9GY29GXG8vJSZhbGlnbkdRJWF4aXNGJy9GX29RKWJhc2VsaW5lRicvRmJvUSdjZW50ZXJGJy9GZG9RJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0Zpbi8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRmlxLyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRmRyLyUmZnJhbWVHRmRyLyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGNC8lLWVxdWFsY29sdW1uc0dGNC8lLWRpc3BsYXlzdHlsZUdGNC8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0Zhci8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJ0Zcby8lKXJlYWRvbmx5R0Zpbi8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRi9GLy9JK21zZW1hbnRpY3NHRiRRKkNvbFZlY3RvckYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRl90LyUrYWN0aW9udHlwZUdRLnJ0YWJsZWFkZHJlc3NGJy1GLDYtUSI9RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSwwLjI3Nzc3NzhlbUYnL0ZFRl91LUZHNiQtRko2KC1GIzYoLUZPNjgtRlI2Jy1GVTYpLUYjNiwtRlo2JVEiY0YnRmduRmpuLUYsNi1RIitGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGZnYtRlo2JVEiYkYnRmduRmpuRmJ2LUZaNiVRImFGJ0ZnbkZqbkZnc0Zcb0Zqc0ZcdEYvRlxvRl5vRmFvRmNvRmVvRmhvRlxvRl5vRmFvRmNvLUZSNictRlU2KS1GIzYsRmh2RmJ2LUkjbW5HRiQ2JFEiMkYnRi9GK0Zbd0Znc0Zcb0Zqc0ZcdEYvRlxvRl5vRmFvRmNvRmVvRmhvRlxvRl5vRmFvRmNvLUZSNictRlU2KS1GIzYlRlt3RmduRmpuRlxvRl5vRmFvRmNvRmVvRmhvRlxvRl5vRmFvRmNvRlxvRlxxRl9xRmFxRmNxRmVxRmdxRmpxRlxyRl9yRmJyRmVyRmdyRmlyRlxzRl5zRmBzRmJzRmVzRmdzRlxvRmpzRlx0Ri9GL0ZfdEZidEZldEZfdEZodC1GWjYjRmBvRlxvRi8=with coefficient matrix which is the inverse of the basis changing matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoQmNoYW5nZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ==we 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Hence the inverse 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 this in terms of the more familiar Cartesian coordinate symbols 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 for the old coefficients and 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 for the new coefficients, we get the following linear coordinate transformation from the old coordinates to the new 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and from the new coordinates to the old coordinates 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 the old and new basis in the coordinate space 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As far as linear operations of function addition and scalar multiplication are concerned, instead of working with the abstract vector space of quadratic functions, we can work entirely in ordinary space using coordinates with respect to the "natural basis" of either space which correspond to each other in this obvious correspondence between the two spaces, or some other basis which might be of interest, like the second one we came up with using the Taylor polynomial. This helps us visualize the abstract 3d vector space in terms of the concrete familiar vector space LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEoJiM4NDc3O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW5HRiQ2JVEiM0YnRjVGNy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNUY3. Visualizable mathematics is always more powerful that symbol pushing.
Here are the old (black) and new (red, blue, green) basis vectors: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Relevance to Linear Differential Equations
An arbitrary quadratic function is the general solution of a simple linear homogeneous differential equation, although Maple uses a slightly different basis of the vector space of solutions, reversing the order of the coefficients back to descending powers, and including a factorial like the Taylor polynomial: 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 solution space of this differential equation is a vector space, and the powers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn provide a natural basis for it. The arbitrary coefficients 1/2 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlEkX0MxRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGMi9GNlEnbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYmUSRfQzJGJ0YvRjJGNUY4LUYsNiZRJF9DM0YnRi9GMkY1RjJGPA== are the corresponding coordinates on that solution space. Thus the machinery we developed for the 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spaces will be important in understanding linear differential equations like this one, whose general solutions are linear combinations of certain linearly independent basic solutions. Once we find enough (linearly independent) basic solutions to span the whole solution space, we get the general solution as an arbitrary linear combination of these basic solutions.Taylor polynomials: Taylor gives us the inverse!The inverse transformation could also obtained by simply evaluating the Taylor polynomial centered at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8 using calculus and comparing coefficients of powers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiUtRiM2Jy1GLDYmUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSgmbWludXM7RidGOi9GPlEnbm9ybWFsRicvJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGPC8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZWLUkjbW5HRiQ2JVEiMUYnRjpGREY6RkRGOkZERitGOkZE on both sides of the equation: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Matching the coefficients on both sides of the equation: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