Homework Sourse1 Let T be a linear transformation of Rn to Rn given by Tv
1. Let T be a linear transformation of Rn to Rn given by T(v) = Av for some n x n matrix A.
Note: ker Tn= nullspace of An
Solution
Post another question to get the answert to the part 3. Thanks, Nikhil
a) T being the linear transformation, where T(v) = Av for some nXn matrix
a) ker Tn = nullspace of An
=> Null space of matrix An where T(1) = A, T(2) = 2A, T(3) = 3A
Hence null space of T1 will be smaller then the null space of T2 and similarly null space of T2 will be smaller than null space of T3
Hence T1 will be a subset of T2 and T2 will be a subset of T3
b) In the given matrix , we have ker T = ker T2
Hence we need to prove that in general that Tn = Ker T(n+1)
let us prove the statement with the help of mathemtical induction.
For the n=1 base case.
Let us assume that n=1, will be implify that T1 = Ker T2, which is true let us asume that now TK = Ker T(K+1)
Now we need to prove that the given statement will be valid for T(k+1)
T(K+1) = Ker T(K+2)
From the second statement substituting n = n+1, we will prove that T(K+1) = T(K+2)
