Let T R4 R3 be defined by Txyst xyst x2st xy3s3t Assume T

Let T: R4 --> R3 be defined by T(x,y,s,t) = (x-y+s+t, x+2s-t, x+y+3s-3t). Assume T is a linear map. Find a linearly independent set of vectors that generates the kernel of T. Hint: Write a system of equations using the definition of the kernel.

Solution

T(x,y,s,t)=0 gives

x-y+s+t=0

x+2s-t=0          ie x=-2s+t

x+y+3s-3t=0

x-y+s+t=-2s+t-y+s+t=0=-y-s+2t

y=-s+2t

x+y+3s-3t=0

-2s+t-s+2t+3s-3t=0

which is true for all t,s

So,(x,y,s,t)=(-2s+t,-s+2t,s,t)=s(-2,-1,1,0)+t(1,2,0,1)

So vectors that generate kernel are

{(-2,-1,1,0),(1,2,0,1)}