To determine: To prove: The function T:V rightarrow W is linear transformation if and only if T(au + bv)=aT(u) + bT(v) for all vectors u and v and all scalars a and b.

Brennan Flores 2020-11-09 Answered
To determine:
To prove:
The function T:VW is linear transformation if and only if
T(au + bv)=aT(u) + bT(v) for all vectors u and v and all scalars a and b.
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Expert Answer

broliY
Answered 2020-11-10 Author has 97 answers
Approach:
The function T is said to be linear transformation if is satisfy the vector addition and scalar multiplication properties.
The linear transformation is given by,
 T(au + bv)=aT(u) + bT(v)  (1)
The vector addition is given by,
T(u + v)=T(u) + T(v)  (2)
The scalar multiplication is given by,
T(cu)=cT(u)  (3)
Calculation:
Let a=1andb=1.
Substitute 1 for a and b in formula (1).
T(1u + 1v)=1T(u) + 1T(v)
=T(u) + T(v)
This shows that the function T is preserves T in vector addition.
Substitute au for cu in formula (3)
T(au)=bT(v)
Substitute au for cu in formula (3)
T(bv)=bT(v)
This shows that the function T is preserves in scalar multiplication.
From above calculation it is clear that the function T preserve the vector addition and scalar multiplication. So, the function T is a linear transformation.
Conclusion:
Thus, the function T:VW is a linear transformation if and only if
T(au + bv)=aT(u) + bT(v)
for all vectors u and v and all scalars a and b.
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