document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168385>Question 1168385</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A)Consider the vector space P2. Define the inner product,<BR>\n" );
document.write( "⟨p, q⟩ =  ∫(from0 to1) p(x)q(x) dx<BR>\n" );
document.write( "Use the Gram-Schmidt process to transform the standard basis S = {1, x, x^2} into an orthonormal basis.\r<BR>\n" );
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document.write( "A.1)Express r(x) = 1 + x + 4x^2 as a linear combination of the vectors in the orthonormal basis for P2 found in the previous exercise.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851576 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=851576\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step by step.\r<BR>\n" );
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document.write( "**A) Gram-Schmidt Process**\r<BR>\n" );
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document.write( "We are given the standard basis S = {1, x, x²} and the inner product ⟨p, q⟩ = ∫(from 0 to 1) p(x)q(x) dx.\r<BR>\n" );
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document.write( "1. **v₁ = u₁ = 1**\r<BR>\n" );
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document.write( "2. **v₂ = u₂ - proj_v₁(u₂) = x - ⟨x, v₁⟩ / ⟨v₁, v₁⟩ * v₁**\r<BR>\n" );
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document.write( "   * ⟨x, v₁⟩ = ∫(from 0 to 1) x * 1 dx = [x²/2] from 0 to 1 = 1/2<BR>\n" );
document.write( "   * ⟨v₁, v₁⟩ = ∫(from 0 to 1) 1 * 1 dx = [x] from 0 to 1 = 1<BR>\n" );
document.write( "   * v₂ = x - (1/2) / 1 * 1 = x - 1/2\r<BR>\n" );
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document.write( "3. **v₃ = u₃ - proj_v₁(u₃) - proj_v₂(u₃) = x² - ⟨x², v₁⟩ / ⟨v₁, v₁⟩ * v₁ - ⟨x², v₂⟩ / ⟨v₂, v₂⟩ * v₂**\r<BR>\n" );
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document.write( "   * ⟨x², v₁⟩ = ∫(from 0 to 1) x² * 1 dx = [x³/3] from 0 to 1 = 1/3<BR>\n" );
document.write( "   * ⟨x², v₂⟩ = ∫(from 0 to 1) x² * (x - 1/2) dx = ∫(from 0 to 1) (x³ - x²/2) dx = [x⁴/4 - x³/6] from 0 to 1 = 1/4 - 1/6 = 1/12<BR>\n" );
document.write( "   * ⟨v₂, v₂⟩ = ∫(from 0 to 1) (x - 1/2)² dx = ∫(from 0 to 1) (x² - x + 1/4) dx = [x³/3 - x²/2 + x/4] from 0 to 1 = 1/3 - 1/2 + 1/4 = 1/12<BR>\n" );
document.write( "   * v₃ = x² - (1/3) / 1 * 1 - (1/12) / (1/12) * (x - 1/2) = x² - 1/3 - (x - 1/2) = x² - x + 1/6\r<BR>\n" );
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document.write( "Now, normalize the vectors:\r<BR>\n" );
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document.write( "1. **e₁ = v₁ / ||v₁||**\r<BR>\n" );
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document.write( "   * ||v₁|| = √⟨v₁, v₁⟩ = √1 = 1<BR>\n" );
document.write( "   * e₁ = 1 / 1 = 1\r<BR>\n" );
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document.write( "2. **e₂ = v₂ / ||v₂||**\r<BR>\n" );
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document.write( "   * ||v₂|| = √⟨v₂, v₂⟩ = √(1/12) = 1 / √12 = 1 / (2√3)<BR>\n" );
document.write( "   * e₂ = (x - 1/2) / (1 / (2√3)) = 2√3(x - 1/2) = 2√3x - √3\r<BR>\n" );
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document.write( "3. **e₃ = v₃ / ||v₃||**\r<BR>\n" );
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document.write( "   * ||v₃|| = √⟨v₃, v₃⟩ = √∫(from 0 to 1) (x² - x + 1/6)² dx = √(1/180) = 1 / √(180) = 1 / (6√5)<BR>\n" );
document.write( "   * e₃ = (x² - x + 1/6) / (1 / (6√5)) = 6√5(x² - x + 1/6) = 6√5x² - 6√5x + √5\r<BR>\n" );
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document.write( "Therefore, the orthonormal basis is {1, 2√3x - √3, 6√5x² - 6√5x + √5}.\r<BR>\n" );
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document.write( "**A.1) Express r(x) = 1 + x + 4x² as a linear combination of the vectors in the orthonormal basis.**\r<BR>\n" );
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document.write( "Let r(x) = c₁e₁ + c₂e₂ + c₃e₃.\r<BR>\n" );
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document.write( "* c₁ = ⟨r(x), e₁⟩ = ∫(from 0 to 1) (1 + x + 4x²) * 1 dx = [x + x²/2 + 4x³/3] from 0 to 1 = 1 + 1/2 + 4/3 = 17/6<BR>\n" );
document.write( "* c₂ = ⟨r(x), e₂⟩ = ∫(from 0 to 1) (1 + x + 4x²) * (2√3x - √3) dx = ∫(from 0 to 1) (2√3x - √3 + 2√3x² - √3x + 8√3x³ - 4√3x²) dx = ∫(from 0 to 1) (8√3x³ - 2√3x² + √3x - √3) dx = [2√3x⁴ - 2√3x³/3 + √3x²/2 - √3x] from 0 to 1 = 2√3 - 2√3/3 + √3/2 - √3 = (12√3 - 4√3 + 3√3 - 6√3) / 6 = 5√3 / 6<BR>\n" );
document.write( "* c₃ = ⟨r(x), e₃⟩ = ∫(from 0 to 1) (1 + x + 4x²) * (6√5x² - 6√5x + √5) dx = ∫(from 0 to 1) (6√5x² - 6√5x + √5 + 6√5x³ - 6√5x² + √5x + 24√5x⁴ - 24√5x³ + 4√5x²) dx = ∫(from 0 to 1) (24√5x⁴ - 18√5x³ + 4√5x² - 5√5x + √5) dx = [24√5x⁵/5 - 18√5x⁴/4 + 4√5x³/3 - 5√5x²/2 + √5x] from 0 to 1 = 24√5/5 - 9√5/2 + 4√5/3 - 5√5/2 + √5 = (144√5 - 135√5 + 40√5 - 75√5 + 30√5) / 30 = 4√5 / 30 = 2√5 / 15\r<BR>\n" );
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document.write( "Therefore, r(x) = (17/6) * 1 + (5√3 / 6) * (2√3x - √3) + (2√5 / 15) * (6√5x² - 6√5x + √5).<BR>\n" );
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