Finding a matrix for a linear transformation

In summary, the conversation is discussing finding the matrix A' for a linear transformation T: R2-->R2 relative to a given basis B' {(1, 0), (1, 1)}. The matrix A' is used to express the transformed vectors T(e1) and T(e2) in terms of the given basis. This is done by finding coefficients a, b, c, and d and putting them into a matrix M. In this way, the matrix A' is used to calculate T(v) for any vector v in R2.
  • #1
'

Homework Statement



Find the matrix A' for T: R2-->R2, where T(x1, x2) = (2x1 - 2x2, -x1 + 3x2), relative to the basis B' {(1, 0), (1, 1)}.

Homework Equations



B' = {(1, 0), (1, 0)} so B'-1 = {(1, -1), (0, 1)}.

The Attempt at a Solution



I'm confused at what exactly a transform matrix relative to a given basis is to mean. Does this mean that some vector vB', when multiplied by A', will equal T(v)B'? I have the solution from the book: B-1AB', but if I had to put in English what it looks to me that this does: B'-1 converts vstd to vB', then A is the transform matrix which should take vstd and output T(v)std, then B' will convert vB' to vstd. It therefore look as if this will take vectors in standard bases and output them in standard bases after transforming? That can't be right because the answer given is not the same as the transform. What is the question asking, exactly?
 
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  • #2
You have to find ##Te_1,Te_2## under the expression of ##e_1,e_2,## where ##\{e_1,e_2\}## is the given basis.
 
  • #3
tommyxu3 said:
You have to find ##Te_1,Te_2## under the expression of ##e_1,e_2,## where ##\{e_1,e_2\}## is the given basis.

Sure, T(e1), T(e2) is easy: {(2, -1), (0, 2)}
 
  • #4
You have to express them ##\textbf{under}## ##e_1,e_2.##
That is, you have to get ##Te_1=ae_1+be_2,Te_2=ce_1+de_2,## likewise.
 
  • #6
I think that if I can get an answer to this then I will be able to figure out what is going on in the answer and explanation in the book:
I'm confused at what exactly a transform matrix relative to a given basis is to mean. Does this mean that some vector vB', when multiplied by A', will equal T(v)B'?
 
  • #7
Take your problem for example. For the given ordered basis ##e_1,e_2,## If I get ##Te_1=ae_1+be_2,Te_2=ce_1+de_2,## and let the matrix ##A## be $$\begin{pmatrix}
a & c \\
b & d
\end{pmatrix}.$$
Then for any vector ##v=xe_1+ye_2,## if you want to get ##Tv,## besides calculating it directly, you can use the matrix like
$$M
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
ax+cy \\
bx+dy
\end{pmatrix}
,$$
which you can notice would be the coefficients of ##Tv,## for ##(ax+cy)e_1+(bx+dy)e_2=x(ae_1+be_2)+y(ce_1+de_2)=xT(e_1)+yT(e_2)=T(xe_1+ye_2)=Tv,## satisfying the rules of linear maps.
 

1. What is a linear transformation?

A linear transformation is a function that maps one vector space to another, while preserving the operations of vector addition and scalar multiplication. In simpler terms, it is a mathematical operation that transforms one set of coordinates into another set of coordinates.

2. How do you find a matrix for a linear transformation?

To find a matrix for a linear transformation, you need to follow these steps:

  1. Choose a basis for the domain and range of the transformation.
  2. Apply the transformation to each basis vector in the domain to get the corresponding output vectors in the range.
  3. Arrange the output vectors in a matrix, with each column representing the coordinates of the output vector.
  4. This matrix is the representation of the linear transformation.

3. Why is finding a matrix for a linear transformation important?

Finding a matrix for a linear transformation is important because it allows us to perform calculations and operations on the transformation using matrix algebra. This can be more efficient and easier to manipulate compared to using the original function or equation of the transformation.

4. Can you have multiple matrices for one linear transformation?

Yes, it is possible to have multiple matrices for one linear transformation. This can happen when there are different choices of bases for the domain and range of the transformation. However, all of these matrices will represent the same linear transformation.

5. How do you determine if a matrix represents a linear transformation?

A matrix represents a linear transformation if and only if it satisfies the properties of a linear transformation, which are:

  1. Preservation of vector addition: T(u + v) = T(u) + T(v) for all vectors u and v in the domain.
  2. Preservation of scalar multiplication: T(cu) = cT(u) for all vector u in the domain and scalar c.

If the matrix satisfies these properties, then it represents a linear transformation.

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