Prove that V is a vector space. V=T(m, n), the set of linear transformations T : R^{m} rightarrow R^{n}, together with the usual addition and scalar multiplication of functions.

ruigE 2020-11-24 Answered
Prove that V is a vector space. V=T(m, n), the set of linear transformations T : Rm  Rn, together with the usual addition and scalar multiplication of functions.
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Expert Answer

Alannej
Answered 2020-11-25 Author has 104 answers

Step 1
V=T(m, n), the set of linear transformations T : Rm  Rn, together with the usual addition and scalar multiplication of functions are clearly defined.
next we need to determine if the five conditions of Definion 7.1 are all met.
If T(u) and T(v) are a linear transformation if for all vectors u and v and Rm, then their sum will also be a linear transformation in Rm, that is, V=T(m, n) is closed under addition.
If T(u) is a linear transformation if for all vectors u in Rm and c is a scalar then i and cT(u) will also be a linear transformation thus V=T(m, n) is also closed under scalar multiplication.
M If T(0)=0 is an identical zero transformation, then if T(u) is in Rm linear transformation, then T(0) + T(u)=T) for all T(u) in Rm, so that T(0) is the zero transformation.
If T(u) linear transformation, it has its inverse linear transformation T(u) so that
T(u) + (T(u))=T(u) + T(u)=0
Step 2
a) We are checking whether you are
v1 + v2=v2 + v1
v1 + v2=T(v1)=T(x1 + x2 +  xm) + T(y1 + y2 +  yn)
=T(y1 + y2 +  yn) + T(x1 + x2 +  xm)=v2 + v1
Therefore, commutativity is valid.
b) Check
(v1 + v2) + v3=v1 + (v2 + v3)
(v1 + v2) + v3=((T(v1) + T(v1)) + T(v3)

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