Give the correct answer and solve the given equation V = T(m, n), The set of lineartrans formations T : R^{m} rightarrow R^{n}, together with the usualadditionand scalar multiplication of functions.

fortdefruitI 2020-11-03 Answered
Give the correct answer and solve the given equation V=T(m, n),
The set of lineartrans formations
T : Rm  Rn,
together with the usualadditionand scalar multiplication of functions.
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Expert Answer

Gennenzip
Answered 2020-11-04 Author has 96 answers
Given that, V=T(m, n),
The set of lineartrans formations
T : Rm  Rn,
together with the usualadditionand scalar multiplication of functions.
Find in the following that V is a vector space.
Since the definition of linear transformation is,
If a function T defined in the following,
T : U  V
satisfy the following properties,
1. T[u1 + u2]=T[u1] + T[u2]
2. T[α u]=α T(u)
for all u1, u2, u in U and alpha is any scalar
Then, T is called linear transformation.
Therefore, using these properties in the given function.
T : Rm  Rn
Assume,
u1, u2, u in U and α is any scalar.
Then
1.T[m1 + m2]=T[m1] + T[m2]
2.T[α m]=α T(m) Therefore, from above shows that all properties of linear transformation satisfy.
So, T(m, n) is vector space with the usual addition and scalar multiplication.
Hence, V is a vector space with the usual addition and scalar multiplication.
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