Let T: R^3→R^2 be given by:
T(x2,x2,x3)= (X1+x2+x3,x2+x3)
Prove that T is a linear transformation. Also find the rank and nullity of T.
T: R3→R2 be given by:
T(x1,x2,x3)=(x1+x2+x3,x2+x3)
First, we need to check T is a linear transformation.
let (x1,x2,x3), (y1,y2,y3) R3 and R
T( (x1,x2,x3) + (y1,y2,y3)) = T((x1,x2,x3) + (y1,y2,y3))
= T( x1+y1, x2+y2, x3+y3)
= (x1+y1+x2+y2+x3+y3 , x2+y2+x3+y3)
= ((x1+x2+x3) + (y1+y2+y3), (x2+x3)+(y2+y3))
= (x1+x2+x3,x2+x3) + (y1+y2+y3,y2+y3)
= T(x1,x2,x3) + T(y1,y2,y3)
since,it is satisfying the definition of linear transformation.
Hence,
T is a linear transformation.
Now we need to find the rank and nullity of T.
as given;
T(x1,x2,x3)=(x1+x2+x3,x2+x3)
here first to find the matrix corresponding to this linear transformation,
let ={(1,0,0),(0,1,0),(0,0,1)}
T(1,0,0) = (1,0) = 1(1,0) + 0(0,1)
T(0,1,0) = (1,1) = 1(1,0) + 1(0,1)
T(0,0,1) = (1,1) = 1(1,0) + 1(0,1)
[T] =
since matrix have two non-zero rows. therefore rank of a matrix is 2.
rank (T)= 2
using rank nullity theorem which states that rank(T) + nullity(T) = n;
rank(T) + nulity(T) = 3 n = 3
nulity(T) = 3 - rank(T) = 3 - 2 =1
nulity(T) = 1
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