could you solve 30 and 31? Starting with the two properties of a linear transfor
ID: 3038638 • Letter: C
Question
could you solve 30 and 31?
Starting with the two properties of a linear transformation T, we found from only Theorem in this Section that T can be computed using a matrix product: T(x^rightarrow) = Ax^rightarrow Use this to prove for any linear transformation T: R^n rightarrow R^n, we must have: T (0^rightarrow_n) =0^rightarrow_m Now, using the Additivity Property and the property of the zero vector, prove directly that for any linear transformation T: R^n rightarrow R^n, we must have (0^rightarrow_n) =0^rightarrow_m.Explanation / Answer
T is a linear transformation
given T (x) = A x
T (On)= A (On) = [ a11 a12 --- a1n
a21 ------------------ a2n
--- --
--- --
an1 --- ann] x (0,0,--------0)T ( 0 repeated n times )
= (0,0,--------0)T ( 0 repeated m times )
= (Om)
ie if T is a LT then identity element of Rn is mapped on to identity element of Rm
2 . Property of LT is T ( u+v) = T(u) +T(v)
ie T ( u+v) = T(u) +T(v) puyt v=0
T( u+0) = T(u) +T(0)
T(u) + Om= T(u) +T(On) => T(On) =Om
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