Let left{v_{1}, v_{2}, dots, v_{n}right} be a basis for a vector space V.

Rivka Thorpe 2021-03-07 Answered

Let {v1, v2,, vn} be a basis for a vector space V. Prove that if a linear transformation T : V V satisfies T(v1)=0 for i=1, 2,s˙, n, then T is the zero transformation.
Getting Started: To prove that T is the zero transformation, you need to show that T(v)=0 for every vector v in V.
(i) Let v be an arbitrary vector in V such that v=c1 v1 + c2 v2 + s˙ + cn vn.
(ii) Use the definition and properties of linear transformations to rewrite T (v) as a linear combination of T (v1).
(iii) Use the fact that T (vi)=0 to conclude that T (v)=0, making T the zero tranformation.

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Answered 2021-03-08 Author has 108 answers

Step 1
Let {v1, v2,, vn} be the basis for a vector space.
Define the map, T : VV such that T (vi)=0 for i=1, 2,s˙, n.
To prove, the above defined map is a linear transformation.
Step 2
Let v1, v2  V and r F a scalar.
 T (v1)=0=T(v2)
As v1 v2 V v1 + v2  V, rv1  V
 T (v1 + v2)=0
=0 + 0
=T (v1) + T (v2)
Also
T (rv1)=0=r.0
 T (rv1)=rT (v1)
Therefore, the above defined map is a linear transformation and this is true for every vector in V.
Step 3
Let v be an arbitrary vector in V such that v=c1 v1 + c2 v2 + s˙ + cn vn
Where, Ci 's are scalars.
The vector v can be written as such linear combination since {v1, v2,, vn} is the basis for a vector space V.
Apply the transformation T on both sides and use the fact that T is linear transformation.
 T (v)=T (c1v1 + c2v2 + s˙ + cnvn)
=T (c1v1) + T(c2v2) + s˙ + T(cnvn)
=c1T (v1) + c2T (v2) + s˙ + cnT (vn)
Also, T
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