Linear Transformation Part 2: Finding the Image of a Linear Transformation

In summary, the linear transformation L maps (1,1) to (1,-2) and (-1,1) to (2,3). To find L(-1,5), we can use the fact that (-1,5) can be written as a linear combination of (1,1) and (-1,1). Similarly, for L(a1,a2), we can write (a1,a2) as a linear combination of (1,1) and (-1,1) and apply the transformation. However, it is important to correctly solve for the coefficients in the linear combination before applying the transformation.
  • #1
aznkid310
109
1

Homework Statement



Let R2 => R2 be a linear transformation for which we know that:

L(1,1) = (1,-2)
L(-1,1) = (2,3)

What is: L(-1,5) and L(a1,a2)?


Homework Equations



I don't know where to start. I tried writing (-1,5) as a linear combo of (1,1) and
(-1,1), but that got me nowhere. Am i suppose to find a basis? How do i do that?

The Attempt at a Solution



(-1,5) = a(1,1) + b(-1,1)

Transforming it into a 2 x 3 matrix and row reducing it, i get a = 2, b = 3.
If i am on the right track, what do i do next?
 
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  • #2
If (-1,5)=2*(1,1)+3*(-1,1), then L(-1,5)=2*L(1,1)+3*L(-1,1), right? That's what 'linear' is all about. Now do the same thing for (a1,a2).
 
  • #3
Ok i think i got part a!

For L(a1,a2), i stuck at: L(a1,a2) = 2(L(a1,a2)) + 3(L(a1,a2))

I was looking at some of examples, and i had no idea where they get the (a+b)/2 and (a-b)/2 terms from.
 
  • #4
Aren't you going to solve (a1,a2)=a*(1,1)+b*(-1,1) first for a and b, so you can do EXACTLY the same thing as you did for part a? Except this time a=2 and b=3 are changed. I think that's the pedagogical point.
 
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  • #5
so this is what i did:

[ 1 -1 a1
1 1 a2 ]

[ 1 -1 a1
0 2 a2 ]

which means b = a2/2, a = a1 + a2/2

Then L(a1,a2) = (a1 + a2/2)( L(1,1)) + (a2/2)(L(-1,2))

= (a1 + a2/2)(1,-2) + (a2/2)(2,3)

= (a1 + a2/2, -2a1 - a2) + (a2, (3a2)/2)

= (a1 + (3a2)/2), (-2a1 + a2/2)

But, this is not the right answer. What did i do wrong?
 
  • #6
aznkid310 said:
so this is what i did:

[ 1 -1 a1
1 1 a2 ]

[ 1 -1 a1
0 2 a2 ]
This is wrong. You subtracted the first row from the second so this should be
[ 1 -1 a1
0 2 a2-a1]
Of course, what you are really saying is that a- b= a1 and a+ b= a2. That should be easy to solve without matrix methods.
which means b = a2/2, a = a1 + a2/2

Then L(a1,a2) = (a1 + a2/2)( L(1,1)) + (a2/2)(L(-1,2))

= (a1 + a2/2)(1,-2) + (a2/2)(2,3)

= (a1 + a2/2, -2a1 - a2) + (a2, (3a2)/2)

= (a1 + (3a2)/2), (-2a1 + a2/2)

But, this is not the right answer. What did i do wrong?
 

1. What is a linear transformation and how is it different from other types of transformations?

A linear transformation is a mathematical operation that maps one vector space to another while preserving its linear structure. This means that the output of a linear transformation is a linear combination of the input vectors. Unlike other types of transformations, a linear transformation also preserves the origin and straight lines, making it a powerful tool in math and science.

2. How is a linear transformation represented?

A linear transformation can be represented by a matrix. Each column of the matrix represents the transformation of a basis vector in the input vector space to a linear combination of the basis vectors in the output vector space. The matrix can be used to perform the transformation on any input vector by matrix multiplication.

3. What is the importance of linear transformations in real-world applications?

Linear transformations are used in a variety of real-world applications, such as computer graphics, image processing, and data analysis. They allow for efficient and accurate manipulation of data and can be used to solve complex problems in fields such as engineering, economics, and physics.

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

To determine if a transformation is linear, you can apply the two properties of linearity: additivity and homogeneity. Additivity means that the transformation of the sum of two vectors is equal to the sum of the individual transformations. Homogeneity means that the transformation of a scalar multiple of a vector is equal to the scalar multiple of the transformation of the vector. If these two properties hold, the transformation is linear.

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

No, a linear transformation cannot change the dimension of the input vector space. The dimension of the output vector space will always be equal to the number of columns in the transformation matrix, which is determined by the number of basis vectors in the input vector space. This means that a linear transformation can only map from a vector space to a vector space with the same dimension or a lower dimension.

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