Linear transformation and Change of Basis

In summary, the conversation is about finding linear transformations T with respect to the bases B and Nat. The first step is to find T with respect to B, which can be done by inspection. To find T with respect to Nat, the natural basis vectors must be written in terms of v_1 and v_2 and then the transformation can be calculated using the linearity of T. One of the participants mentions using the terms e_1 and e_2 instead of the natural basis vectors.
  • #1
Jimmy84
191
0

Homework Statement



Greetings, I have been stuck with this problem for a while, I thought maybe someone could give me some advice about it. Thanks a lot in advance.

If T is a linear transformation that goes from R^2 to R^2 given that T(v1)= -2v2 -v1 and
T(v2)=3v2.

and B = v1=(1,1) , v2=(1,-1)


Find T with respect to the base B and T with respect to Nat, (the Natural Base)

Homework Equations





The Attempt at a Solution



I found T with respect to B by inspection

-1 0
-2 3

How can I find T with respect to the natural base?

Thanks
 
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  • #2
Write the two natural basis vectors in terms of [itex]v_1,v_2[/itex] and then see what this transformation does to them. Once you know that, then you know how to find the matrix.
 
  • #3
Robert1986 said:
Write the two natural basis vectors in terms of [itex]v_1,v_2[/itex] and then see what this transformation does to them. Once you know that, then you know how to find the matrix.

Do you mean to write 1,0 and 0,1 as a linear combination of v1 and v2 ? how can I see what the transformation does to them when I'm not given the transformation?

Thanks
 
  • #4
Well, let's say [itex]e_1 = c_1v_1 + c_2v_2[/itex]. Then, [itex]T(e_1) = T(c_1v_1 + c_2v_2)[/itex]. Now, use the linearity of [itex]T[/itex] and what you know about [itex]T(v_1)[/itex] and [itex]T(v_2)[/itex] to calculate [itex]T(e_1)[/itex].

EDIT:
I don't know what terms your book uses, but I mean that [itex]e_1 = (1,0)[/itex].
 

Related to Linear transformation and Change of Basis

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another while preserving the operations of vector addition and scalar multiplication. In other words, it is a function that takes in vectors as inputs and produces new vectors as outputs, while maintaining the same structure and properties.

2. How are linear transformations represented?

Linear transformations can be represented in various ways, such as matrices, equations, or graphs. In the case of matrices, a linear transformation can be represented by a square matrix that maps the original vector to the transformed vector. Equations can also be used to represent linear transformations, where each variable represents a different component of the vector. Lastly, graphs can visually show the effect of a linear transformation on a vector.

3. What is the difference between a linear transformation and a change of basis?

A linear transformation is a function that maps one vector space to another, while a change of basis is a way to represent the same vector in a different coordinate system. In other words, a linear transformation changes the vector itself, while a change of basis changes the way we describe the vector.

4. How do you determine if a linear transformation is invertible?

A linear transformation is invertible if and only if it has a unique inverse. In other words, the transformation must be one-to-one (injective) and onto (surjective). This means that every output vector must have a unique input vector, and every possible output vector must be mapped to by at least one input vector.

5. How do you perform a change of basis?

To perform a change of basis, you first need to determine the transformation matrix between the original basis and the new basis. This can be done by writing the new basis vectors in terms of the original basis vectors and then arranging these vectors as columns in a matrix. Then, to change the basis of a vector, you multiply the transformation matrix by the original vector. This will give you the vector represented in the new basis.

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