Let beta = (x^{2} - x, x^{2} + 1,x - 1), beta' = (x^2 - 2x - 3, - 2x^2 + 5x + 5, 2x^2 - x - 3) be ordered bases for P_2(C). Find the change of coordinate matrix Q that changes beta ' -coordinates into beta -coordinates.

Question
Alternate coordinate systems
asked 2021-02-08
Let \(\displaystyle\beta={\left({x}^{{{2}}}-{x},{x}^{{{2}}}+{1},{x}-{1}\right)},\beta'={\left({x}^{{2}}-{2}{x}-{3},-{2}{x}^{{2}}+{5}{x}+{5},{2}{x}^{{2}}-{x}-{3}\right)}\) be ordered bases for \(\displaystyle{P}_{{2}}{\left({C}\right)}.\) Find the change of coordinate matrix Q that changes \(\displaystyle\beta'\) -coordinates into \(\displaystyle\beta\) -coordinates.

Answers (1)

2021-02-09
LET A = \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{1}&{1}&{0}&{1}&-{2}&{2}\backslash{h}{l}\in{e}-{1}&{0}&{1}&-{2}&{5}&-{1}\backslash{h}{l}\in{e}{0}&{1}&-{1}&-{3}&{5}&-{3}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\) It may be observed that the entries in the columns of A are the coefficients of x2, x and the scalar multiples of 1 in the ordered bases β and β’ respectively. To determine the change of coordinates matrix from \(\displaystyle\beta’\to\beta,\) we will reduce A to its RREF as under: Add 1 times the 1st row to the 2nd row Add -1 times the 2nd row to the 3rd row Multiply the 3rd row by -1/2 Add -1 times the 3rd row to the 2nd row Add -1 times the 2nd row to the 1st row The RREF of A is \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{1}&{0}&{0}&{3}&-{6}&{3}\backslash{h}{l}\in{e}{0}&{1}&{0}&-{2}&{4}&-{1}\backslash{h}{l}\in{e}{0}&{0}&{1}&{1}&-{1}&{2}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\) Then the change of coordinates matrix from \beta’ to \beta is Q= \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{3}&-{6}&{3}\backslash{h}{l}\in{e}-{2}&{4}&-{1}\backslash{h}{l}\in{e}{1}&-{1}&{2}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
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