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anonymous

  • 5 years ago

y₁ Let U:= { y₂ : y₁=y₂=y₃+y₄ } y₃ y₄ a) Show U is a subspace of F⁴ b) Find a basis for U and prove that it is in fact a basis

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  1. watchmath
    • 5 years ago
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    Is that y1+y2=y3=y4 ?

  2. watchmath
    • 5 years ago
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    I mean y1+y2=y3+y4?

  3. anonymous
    • 5 years ago
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    No its y1=y2=y3+y4

  4. watchmath
    • 5 years ago
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    Define a linear map \(T:F^4\rightarrow F^2\) given by \(T(y_1,y_2,y_3,y_4)=(y_1-y_2,y_2-y_3-y_4)\). Notice that \(\ker t\) is the collection of \(((y_1,y_2,y_3,y_4)^T\) such that \(y_1-y_2=0\) and \(y_2-y_3-y_4=0\). Hence \(\ker T=U\). Therefore \(U\) is a subspace of (F^4\). We claim that \(B=\{(1,1,0,1)^T,(1,1,1,0)^T\}\) is a basis of \(U\). One can easily check that each \(B\subset U\). Suppose \(s(1,1,0,1)^T+t(1,1,1,0)=0\) Then \((s+t,s+t,t,s)=(0,0,0,0)\) By looking at the last two coordinates, we have \(s=t=0\). Hence \(B\) is linearly independent. You just need to prove that \(B\) spans \(U\) (give it a try! :) )

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