document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182632>Question 1182632</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the representation matrix of a linear transformation over the basis B, how to find a basis for a different representation matrix for the same linear transformation in which a certain condition is required? <BR>\n" );
document.write( "For example: \r<BR>\n" );
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document.write( "V = R2[x] (all the polynomials of power 2 or less over Real numbers)\r<BR>\n" );
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document.write( "Given the following basis:<BR>\n" );
document.write( "B= {1, 1+x, x^2 -1}\r<BR>\n" );
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document.write( "Defining T: V -> V a linear transformation as following:<BR>\n" );
document.write( "T(1) = 4 - 3x^2<BR>\n" );
document.write( "T(1+x) = 4 + 4x + 3x^2<BR>\n" );
document.write( "T(x^2 -1) = x^2 - 4\r<BR>\n" );
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document.write( "The representative matrix of T is:<BR>\n" );
document.write( "1     3   -3<BR>\n" );
document.write( "0     4    0    =    [T]B<BR>\n" );
document.write( "-3    3    1\r<BR>\n" );
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document.write( "If given that T is diagonalizable, and the eigenvalues of it are { 4, 4, -2}\r<BR>\n" );
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document.write( "Find a basis of V for which the representation matrix is a diagonal matrix. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849916 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849916\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find a basis for which the representation matrix of T is diagonal, given that T is diagonalizable:\r<BR>\n" );
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document.write( "**1. Eigenvalues and Eigenvectors:**\r<BR>\n" );
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document.write( "You're given the eigenvalues: λ₁ = 4, λ₂ = 4 (repeated), and λ₃ = -2.  Since T is diagonalizable, there must be two linearly independent eigenvectors associated with the eigenvalue λ = 4.\r<BR>\n" );
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document.write( "* **For λ = 4:** Solve the equation (T - 4I)v = 0, where v is the eigenvector.  In matrix form, using the given representation matrix [T]B:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "[ 1-4   3  -3 ] [x]   [0]<BR>\n" );
document.write( "[  0   4-4   0 ] [y] = [0]<BR>\n" );
document.write( "[ -3    3  1-4 ] [z]   [0]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "[ -3   3  -3 ] [x]   [0]<BR>\n" );
document.write( "[  0   0   0 ] [y] = [0]<BR>\n" );
document.write( "[ -3   3  -3 ] [z]   [0]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "This simplifies to -3x + 3y - 3z = 0, or x = y - z.  We have two free variables.  Let z = t and y = s.  Then x = s - t.  The eigenvectors associated with λ = 4 can be written as:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "[ s - t ]<BR>\n" );
document.write( "[   s   ]<BR>\n" );
document.write( "[   t   ]  =  s[1] + t[-1]<BR>\n" );
document.write( "[ 1 ]     [ 0 ]<BR>\n" );
document.write( "[ 0 ]     [ 1 ]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "So, two linearly independent eigenvectors corresponding to λ = 4 (in the B basis) are [1, 1, 0]ᵀ and [-1, 0, 1]ᵀ.\r<BR>\n" );
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document.write( "* **For λ = -2:** Solve (T + 2I)v = 0:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "[ 1+2   3  -3 ] [x]   [0]<BR>\n" );
document.write( "[  0   4+2   0 ] [y] = [0]<BR>\n" );
document.write( "[ -3    3  1+2 ] [z]   [0]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "[ 3   3  -3 ] [x]   [0]<BR>\n" );
document.write( "[ 0   6   0 ] [y] = [0]<BR>\n" );
document.write( "[ -3  3   3 ] [z]   [0]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "From this, we get y = 0 and 3x - 3z = 0, or x = z. Let x = t, then z = t. The eigenvector is:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "[ t ]<BR>\n" );
document.write( "[ 0 ]<BR>\n" );
document.write( "[ t ] = t [ 1 ]<BR>\n" );
document.write( "[ 0 ]<BR>\n" );
document.write( "[ 1 ]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "So, an eigenvector corresponding to λ = -2 (in the B basis) is [1, 0, 1]ᵀ.\r<BR>\n" );
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document.write( "**2. Change of Basis:**\r<BR>\n" );
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document.write( "Now we have the eigenvectors in the B basis.  We need to convert them to the standard basis (or any other basis you prefer) to get the eigenvectors in terms of polynomials.  Recall that B = {1, 1+x, x²-1}.\r<BR>\n" );
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document.write( "* **Eigenvector [1, 1, 0]ᵀ (for λ = 4):**<BR>\n" );
document.write( "   1*(1) + 1*(1+x) + 0*(x²-1) = 1 + 1 + x = 2 + x\r<BR>\n" );
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document.write( "* **Eigenvector [-1, 0, 1]ᵀ (for λ = 4):**<BR>\n" );
document.write( "   -1*(1) + 0*(1+x) + 1*(x²-1) = -1 + x² - 1 = x² - 2\r<BR>\n" );
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document.write( "* **Eigenvector [1, 0, 1]ᵀ (for λ = -2):**<BR>\n" );
document.write( "   1*(1) + 0*(1+x) + 1*(x²-1) = 1 + x² - 1 = x²\r<BR>\n" );
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document.write( "**3. Diagonalizing Basis:**\r<BR>\n" );
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document.write( "The desired basis for which T is represented by a diagonal matrix is the set of eigenvectors we just found:\r<BR>\n" );
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document.write( "{2 + x, x² - 2, x²}\r<BR>\n" );
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document.write( "In this basis, the matrix representation of T will be diagonal, with the eigenvalues on the diagonal:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "[ 4  0  0 ]<BR>\n" );
document.write( "[ 0  4  0 ]<BR>\n" );
document.write( "[ 0  0 -2 ]<BR>\n" );
document.write( "```<BR>\n" );
document.write( " </SPAN>\n" );
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