Question

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

Answer 1

Is that y1+y2=y3=y4 ?

Answer 2

I mean y1+y2=y3+y4?

Answer 3

No its y1=y2=y3+y4

Answer 4

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! :) )