Linear transformation, basis.

In summary, a linear transformation is a mathematical function that preserves the linear structure of a vector space by systematically changing the coordinates of a vector. A basis in linear algebra is a set of linearly independent vectors that can be used to uniquely represent any vector in a given space. To determine if a set of vectors is a basis, they must be linearly independent and span the entire vector space. Linear transformations can change the dimension of a vector space, and a basis plays a crucial role in understanding how the transformation affects the vectors in the space.
  • #1
pyroknife
613
3
Suppose that T1: V → V and T2: V → V are
linear operators and {v1, . . . , vn} is a basis for V .
If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n, show
that T1(v) = T2(v) for all v in V .


I don't understand this question.
They said If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n
wouldn't that mean T1(v)=T2(v) already? I don't get what I have to prove here.


Isn't this just saying
T1(v1)=T2(v1)
T1(v2)=T2(v2)
.
.
.
T1(vn)=T2(vn)?
 
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  • #2
pyroknife said:
Suppose that T1: V → V and T2: V → V are
linear operators and {v1, . . . , vn} is a basis for V .
If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n, show
that T1(v) = T2(v) for all v in V .


I don't understand this question.
They said If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n
wouldn't that mean T1(v)=T2(v) already? I don't get what I have to prove here.


Isn't this just saying
T1(v1)=T2(v1)
T1(v2)=T2(v2)
.
.
.
T1(vn)=T2(vn)?

No. You are given that T1 and T2 give the same thing for the basis vectors. What about any other vector v? Show T1(v) = T2(v).
 
  • #3
T1(v) = T1(c1v1 + ... + cnvn)
= T1(c1v1) + ... + T1(cnvn)
= c1T1(v1) + ... + cnT1(vn)
= c1T2(v1) + ... + cnT2(vn)
= T2(c1v1) + ... + T2(cnvn)
= T2(c1v1 + ... + cnvn)
= T2(v)
 
  • #5
pyroknife said:
T1(v) = T1(c1v1 + ... + cnvn)
= T1(c1v1) + ... + T1(cnvn)
= c1T1(v1) + ... + cnT1(vn)
= c1T2(v1) + ... + cnT2(vn)
= T2(c1v1) + ... + T2(cnvn)
= T2(c1v1 + ... + cnvn)
= T2(v)
Looks good. The only thing I would add is a statement like "Let v be an arbitrary member of V" or "For all v in V, we have..."

Edit: D'oh...too slow.
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the linear structure of the original space. In simpler terms, it is a way of transforming or changing the coordinates of a vector in a systematic way.

2. What is a basis in linear algebra?

A basis is a set of linearly independent vectors that span a vector space. This means that any vector in the space can be written as a linear combination of the basis vectors. In other words, a basis is a set of vectors that can be used to uniquely represent any vector in a given vector space.

3. How do you determine if a set of vectors is a basis?

To determine if a set of vectors is a basis, you can check two conditions. First, the vectors must be linearly independent, meaning that none of the vectors can be written as a linear combination of the others. Second, the vectors must span the entire vector space, meaning that any vector in the space can be written as a linear combination of the basis vectors.

4. Can a linear transformation change the dimension of a vector space?

Yes, a linear transformation can change the dimension of a vector space. For example, a 2D vector can be transformed into a 3D vector through a linear transformation. This is because a linear transformation can change the coordinates of a vector in a systematic way, which can result in a different dimension.

5. What is the role of a basis in a linear transformation?

A basis plays a crucial role in a linear transformation as it provides a framework for understanding how the transformation affects the vectors in a vector space. By using a basis, we can analyze the transformation by looking at how it affects the basis vectors and use that information to understand its effect on any vector in the space.

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