Linear transformation representation with a matrix

In summary: To get the actual vector in B2 we simply express v in terms of B2 and multiply.In summary, to find the transformation of a vector expressed in a nonstandard basis, we first transform the basis vectors to the standard basis, then express the vector in terms of the transformed basis, and finally multiply by the transformation matrix to get the vector in the nonstandard basis.
  • #1

Homework Statement



For the linear transformation T: R2-->R2 defined by T(x1, X2) = (x1 + x2, 2x1 - x2), use the matrix A to find T(v), where v = (2, 1). B = {(1, 2), (-1, 1)} and B' = {(1, 0), (0, 1)}.

Homework Equations



T(v) is given, (x1+x2, 2x1-x2)

The Attempt at a Solution



Okay, I see that T(v) is simply (2+1, 2*2-1) --> (3, 3), but this matrix business has me a bit confused. From my textbook:

Using the basis B = {(1, 2), (-1, 1)}, you find that v = (2, 1) = 1(1, 2) - 1(-1, 1), which implies [v]B = [1 -1]T.

Now, where did this seemingly 'magical' 1 and -1 come from? The matrix relative to B and B' is obviously {(3, 0), (0, 3)}, but I have no idea where this [1 -1]T came from, nor what it is. I see that this is the coordinate matrix for (3, 3) relative to B, because I see that A*v in this case gives (3, 3), but have no idea how it was derived. Perhaps there's some small gem or concept here that I'm missing?
 
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  • #2
patricio2626 said:

Homework Statement



For the linear transformation T: R2-->R2 defined by T(x1, X2) = (x1 + x2, 2x1 - x2), use the matrix A to find T(v), where v = (2, 1). B = {(1, 2), (-1, 1)} and B' = {(1, 0), (0, 1)}.

Homework Equations



T(v) is given, (x1+x2, 2x1-x2)

The Attempt at a Solution



Okay, I see that T(v) is simply (2+1, 2*2-1) --> (3, 3), but this matrix business has me a bit confused. From my textbook:

Using the basis B = {(1, 2), (-1, 1)}, you find that v = (2, 1) = 1(1, 2) - 1(-1, 1), which implies [v]B = [1 -1]T.

Now, where did this seemingly 'magical' 1 and -1 come from? The matrix relative to B and B' is obviously {(3, 0), (0, -3)}, but I have no idea where this [1 -1]T came from, nor what it is. I see that this is the coordinate matrix for (3, 3) relative to B, because I see that A*v in this case gives (3, 3), but have no idea how it was derived. Perhaps there's some small gem or concept here that I'm missing?
The coordinates of a vector represent the constants that multiply the vectors in a basis.
In the standard basis for ##\mathbb{R}^2##, the vector ##\begin{bmatrix} 3 \\ 2\end{bmatrix}## means ##3\begin{bmatrix} 1 \\ 0\end{bmatrix} + 2\begin{bmatrix} 0\\ 1\end{bmatrix}##.
What you show as ##[v]_B## are the coordinates of basis B to result in ##\begin{bmatrix} 2 \\ 1 \end{bmatrix}##
 
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  • #3
Mark44 said:
The coordinates of a vector represent the constants that multiply the vectors in a basis.
In the standard basis for ##\mathbb{R}^2##, the vector ##\begin{bmatrix} 3 \\ 2\end{bmatrix}## means ##3\begin{bmatrix} 1 \\ 0\end{bmatrix} + 2\begin{bmatrix} 0\\ 1\end{bmatrix}##.
What you show as ##[v]_B## are the coordinates of basis B to result in ##\begin{bmatrix} 2 \\ 1 \end{bmatrix}##

Okay, so

v = (2, 1) = 1(1, 2) - 1(-1, 1)

because
1*1 - 1*-1 = 2
1*2 - 1*1 = 1

So, the book simply skipped this piece of the explanation?
 
  • #4
patricio2626 said:
Okay, so

v = (2, 1) = 1(1, 2) - 1(-1, 1)

because
1*1 - 1*-1 = 2
1*2 - 1*1 = 1

So, the book simply skipped this piece of the explanation?

Yes.
 
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  • #5
Thanks gents, this got me past this headscratcher, and then I had an on-the-fly tutor session yesterday and I get it now. To convert between nonstandard bases and find a transform for a vector expressed in the standard basis:

-To get the relative matrix we transform the first nonstandard base, say, B1, and express each column vector as a linear combination of our second nonstandard base, say, B2
-To get v expressed in terms of multiples of B1 we express v as a linear combination of B1.
-We then multiply the result by our transform matrix: Av, and this is expressed in terms of our coefficients for B2
 

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 vector operations of addition and scalar multiplication. In simpler terms, it is a transformation that preserves straight lines and the origin.

2. How is a linear transformation represented with a matrix?

A linear transformation can be represented with a matrix by using the coordinates of the input and output vectors. The columns of the matrix represent the images of the basis vectors from the input space, and the linear combination of these column vectors represents the image of any input vector.

3. What is the purpose of representing a linear transformation with a matrix?

Representing a linear transformation with a matrix provides a convenient and efficient way to perform calculations and transformations on vectors. It also allows for the use of matrix algebra to solve problems involving linear transformations.

4. How do you determine the matrix representation of a linear transformation?

To determine the matrix representation of a linear transformation, you first need to choose a basis for both the input and output vector spaces. Then, you can find the images of the basis vectors in the output space and use those images to construct a matrix that represents the transformation.

5. Can any linear transformation be represented with a matrix?

Yes, any linear transformation can be represented with a matrix as long as the input and output vector spaces have a finite number of dimensions. However, the matrix representation may not be unique as it depends on the choice of basis vectors for the vector spaces.

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