## 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

1. watchmath

Is that y1+y2=y3=y4 ?

2. watchmath

I mean y1+y2=y3+y4?

3. anonymous

No its y1=y2=y3+y4

4. watchmath

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! :) )