The equations you provided are vector-valued and Sage expects scalar ones
Is that what you are expecting:
a,b,c,d,e,f,t = var('a,b,c,d,e,f,t')
v(t)=(t,t)
u(t)=(2+t,1-t)
B(t)=(a*t^2+b*t+c,d*t^2+e*t+f)
eq10=v(t).substitute(t==1)[0]==B(t).substitute(t==0)[0] # v(1)==B(0)
eq11=v(t).substitute(t==1)[1]==B(t).substitute(t==0)[1] # v(1)==B(0)
eq20=u(t).substitute(t==0)[0]==B(t).substitute(t==1)[0] # u(0)==B(1)
eq21=u(t).substitute(t==0)[1]==B(t).substitute(t==1)[1] # u(0)==B(1)
eq30=diff(v(t),t).substitute(t==1)[0]==diff(B(t),t).substitute(t==0)[0] #
v'(1)=B'(0)
eq31=diff(v(t),t).substitute(t==1)[1]==diff(B(t),t).substitute(t==0)[1] #
v'(1)=B'(0)
eq40=diff(u(t),t).substitute(t==0)[0]==diff(B(t),t).substitute(t==1)[0] #
u'(0)=B'(1)
eq41=diff(u(t),t).substitute(t==0)[1]==diff(B(t),t).substitute(t==1)[1] #
u'(0)=B'(1)
solve([eq10,eq11,eq20,eq21,eq30,eq31,eq40,eq41],a,b,c,d,e,f,t)
#[[a == 0, b == 1, c == 1, d == -1, e == 1, f == 1, t == r1]]
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