document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205688>Question 1205688</A>: <I><SPAN STYLE=\"word-break: break-all;\">  If T: R³→ R³ is If is a linear transformation defined by T(x,y,z)=(x+2y,x−y+z,−2y+z) Write the primary decomposition of R³ </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847954 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=847954\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the primary decomposition of R³ under the linear transformation T, we need to find the eigenvalues and eigenvectors of the matrix associated with T. \r<BR>\n" );
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document.write( "**Step 1: Find the Matrix Representation of T**\r<BR>\n" );
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document.write( "The matrix representation of T is:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "A = [[1, 2, 0],<BR>\n" );
document.write( "     [1, -1, 1],<BR>\n" );
document.write( "     [0, -2, 1]]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Step 2: Find the Eigenvalues of A**\r<BR>\n" );
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document.write( "To find the eigenvalues, we solve the characteristic equation det(A - λI) = 0:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "det([[1-λ, 2, 0],<BR>\n" );
document.write( "     [1, -1-λ, 1],<BR>\n" );
document.write( "     [0, -2, 1-λ]]) = 0<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Solving this equation, we find the eigenvalues: λ₁ = 1, λ₂ = -1, and λ₃ = 1.\r<BR>\n" );
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document.write( "**Step 3: Find the Eigenspaces**\r<BR>\n" );
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document.write( "For each eigenvalue, we find the corresponding eigenspace:\r<BR>\n" );
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document.write( "* **For λ₁ = 1:**<BR>\n" );
document.write( "  Solve the equation (A - I)v = 0:<BR>\n" );
document.write( "  ```<BR>\n" );
document.write( "  [[0, 2, 0],<BR>\n" );
document.write( "   [1, -2, 1],<BR>\n" );
document.write( "   [0, -2, 0]]v = 0<BR>\n" );
document.write( "  ```<BR>\n" );
document.write( "  The eigenspace E₁ is spanned by {(1, 0, 1), (0, 1, 1)}.\r<BR>\n" );
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document.write( "* **For λ₂ = -1:**<BR>\n" );
document.write( "  Solve the equation (A + I)v = 0:<BR>\n" );
document.write( "  ```<BR>\n" );
document.write( "  [[2, 2, 0],<BR>\n" );
document.write( "   [1, 0, 1],<BR>\n" );
document.write( "   [0, -2, 2]]v = 0<BR>\n" );
document.write( "  ```<BR>\n" );
document.write( "  The eigenspace E₂ is spanned by {(1, -1, 1)}.\r<BR>\n" );
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document.write( "**Step 4: Primary Decomposition**\r<BR>\n" );
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document.write( "Since the eigenvalues are distinct, the primary decomposition of R³ is the direct sum of the eigenspaces:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "R³ = E₁ ⊕ E₂<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "where:<BR>\n" );
document.write( "* E₁ = span{(1, 0, 1), (0, 1, 1)}<BR>\n" );
document.write( "* E₂ = span{(1, -1, 1)}\r<BR>\n" );
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document.write( "This means that every vector in R³ can be uniquely expressed as a sum of a vector in E₁ and a vector in E₂.<BR>\n" );
document.write( " </SPAN>\n" );
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