Another linear algebra problem, basis and linear transformations.

In summary: The matrix T is in the standard basis for R3 and R2. It can be found by multiplying a vector in alternate coordinates by the basis vectors and converting it to standard coordinates. The inverse of this matrix will take standard coordinates to alternate coordinates.
  • #1
Gramsci
66
0

Homework Statement


The matrix A =(1,2,3;4 5 6) defines a linear transformation T: R^3-->R^2 . Find the transformation matrix for T with respect to the basis (1,0,1),(0,2,0),(-1,0,1) for R^3 and the basis (0,1),(1,0) for R^2.

Homework Equations


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The Attempt at a Solution


I have no idea really. How do I change basis here when I have to basis?
 
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  • #2
The matrix you have is a representation of an abstract linear transformation RELATIVE to a specific basis. It's domain and range spaces will both have bases which are indicated by its form. A good starting point would be to examine how that transformation acts on the basis vectors of the second set.
 
  • #3
I'm not sure I understand. What set?
 
  • #4
Gramsci said:
I'm not sure I understand. What set?

the basis vectors form a set. {v1, v1, ..., vn} I meant how the transformation acts on {vi} you want to see how the transformation acts on the second set of basis vectors.
 
  • #5
Ah, it transforms it. I don't see where this is leading me, however :/
 
  • #6
Gramsci said:
Ah, it transforms it. I don't see where this is leading me, however :/

Does the question specify the representation of T is in the standard bases for R3 and R2?
 
  • #7
yes,it does.
 
  • #8
The the general procedure is as follows:
you are given bases, which can be though of as rules transforming a vector from one coordinate system to another. the list of basis elements you are given are the basis vectors of your space written in the standard coordinate system. If you multiply a vector in you 'alternate' coordinates by your basis vectors (as a matrix they can be the columns) you will then convert it to a vector in standard coordinates. In this case you want to take a vector in standard coordinates and convert it to a vector in alternate coordinates.

If {(1,0,1)T, (0,2,0)T, (-1,0,1)T} convert 'alternate vectors' into 'standard vectors' then the inverse of this matrix will take 'standard vectors' to alternate vectors. and likewise for the range space. The transformation matrix (given in standard coordinates) must then be transformed as well, given the inverse elements dimensions you will hopefully see the way to do this.

you have a few elements to look at: Te (that is, T in the standard basis), Peb ={(1,0,1)T, (0,2,0)T, (-1,0,1)T} (take these as the columns of a 3x3 matrix and Beb ={(0,1)T, (1,0)T}. What you want is Pbe and Bbe. Fortunately these are the inverses of Peb and Beb. Hopefully you have it from here!
 

1. What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that span a vector space. This means that any vector in the vector space can be written as a linear combination of the basis vectors.

2. How do you determine if a set of vectors form a basis?

To determine if a set of vectors form a basis, you can use the rank-nullity theorem. If the dimension of the vector space is equal to the number of vectors in the set and the set is linearly independent, then the set forms a basis.

3. What is a linear transformation?

A linear transformation is a function that maps one vector space to another, while preserving the vector operations of addition and scalar multiplication. In other words, the output of a linear transformation is always a linear combination of the inputs.

4. How do you represent a linear transformation in matrix form?

A linear transformation can be represented in matrix form by using the standard basis vectors as columns in the matrix. The resulting matrix is called the transformation matrix and can be used to perform the transformation on any vector in the vector space.

5. What is the relationship between a basis and a linear transformation?

A basis is essential in defining a linear transformation, as it determines the domain and range of the transformation. The basis vectors can also be used to represent the transformation in matrix form, making it easier to perform calculations and understand the behavior of the transformation.

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