Guided Proof Let {v_{1}, v_{2}, .... V_{n}} be a basis for a vector space V. Prove that if a linear transformation T : V rightarrow V satisfies T (v_{

a)Given:
The linear transformation $T:V\to V$
represented as $T(v_i)=0fori=1,2,...,n.$
Approach:
Consider an arbitrary $v=v_1,v_2,...,v_n$ is basis for V.
The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.
The linear transformation is given by,
$T(v_i)=0,i=1,2,\cdots ,n...(1)$
Calculation:
As the vector set v is the subspace of V, the vector v can be written linear combination.
Write the subspace vas linear combination.
$v=c_1{v}_1+c_2{v}_2+\cdots +c_n{v}_n\cdots ,(2)$
Here, $c_1,c_2,\cdots c_n$ are arbitrary scalars.
Conclusion:
Hence, it is proved above that the set $(v_1,v_2,\cdots v_n)$ is represented as
$v=c_1{n}_1+c_2{v}_2,+\cdots +c_n{v}_n.$
b)Given:
The linear transformation $T:V\to V$
represented as $T(v_i)=0fori=1,2,\cdots ,n.$
Approach:
Consider an arbitrary $v=v_1,v_2,\cdots ,v_n$ is basis for V.
The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.
The linear transformation is given by,
$T(v_i)=0,i=1,2,\cdots ,n\cdots (1)$
The vector additinon is given by,
$T(u+v)=T(u)+T(v)$
The scalar multiplication is given by,
$T(cu)=cT(u)$
Calculation:
As the vector set v is the subspace of V, the vector v can be written linear combination.
Write the subspace vas linear combination.
$T(v)=(c_1{v}_1+c_2{v}_2+\cdots +c_n{v}_n)$
$=T(c_1{v}_1)+T(c_2{v}_2)+\cdots +T(c_n{v}_n)$
$=c_{1}T(v_1)+c_{2}T(v_2)+\cdots +c_{n}T(v_n)....(3)$
Conclusion:
The transformation form of linear combination $v=c_1{v}_1+c_2{v}_2+\cdots +c_n{v}_n$ is
$T(v)=c_{1}T(v_1)+c_{2}T(v_2)+\cdots +c_{n}T(v_n)$
c) Given:
The linear transformation $T:V\to V$
represented as $T(v_i)=0fori=1,2,\cdots ,n.$
Approach:
Consider an arbitrary $v=v_1,v_2,\cdots ,v_n$ is basis for V.
The function T is said to be linear transformation if it satisfies the vector addition and scalar multiplication properties.
The linear transformation is given by,
$T(v_i)=0,i=1,2,\cdots ,n\cdots (1)$
The vector additinon is given by,
$T(u+v)=T(u)+T(v)$
The scalar multiplication is given by,
$T(cu)=cT(u)$
Calculation:
Solve formula (3) with use of formula(1)
$T(v)=c_{1}T(v_1)+c_{2}T(v_2)+\cdots +c_{n}T(v_n)$
$=c_1(0)+c_2(0)+\cdots +c_n(0)$
= 0
From above calculation is is clear linear transformation $T:V\to V$
satisfies