 # 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 $T:V\to W$ is linear transformation if and only if
for all vectors u and v and all scalars a and b.
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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,

The vector addition is given by,

The scalar multiplication is given by,

Calculation:
Let $a=1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}b=1.$
Substitute 1 for a and b in formula (1).

This shows that the function T is preserves T in vector addition.
Substitute au for cu in formula (3)
$T\left(au\right)=bT\left(v\right)$
Substitute au for cu in formula (3)
$T\left(bv\right)=bT\left(v\right)$
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 $T:V\to W$ is a linear transformation if and only if

for all vectors u and v and all scalars a and b.