Linear algebra: transformations

In summary, the conversation discusses showing that a composite linear transformation L = L2 * L1 is also a linear transformation by proving the properties of linearity. It is concluded that the proof is on the right track and it is only necessary to note that L2 and L1 are both linear transformations to complete the proof.
  • #1
seang
184
0
Hello, I'll be online until I get this one completely figured out, so baby steps are for the win here.

Let L1:U->V and L2:U->W be linear transformations, and let L = L2 * L1 be the mapping defined by:

L(u) = L2(L1(u))

for each u which lies in U. Show that L is a linear transformation mapping U into W.
So basically, should I first show that L1(u) is a valid linear transform?, and then show that L2, is, too?
 
Last edited:
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  • #2
Hm, I believe it's enough to assume L1 and L2 are linear transformations, and then proove that L is one, too.
 
  • #3
Is L2 from U to W or from V to W?
 
  • #4
V to W! thank you for that!
What about this? (in progress (latex noobie))

[tex]L(u) = L_2 (L_1 (u) )}[/tex]

[tex]L(u_1 + u_2) = L_2 (L_1 (u_1 + u_2) )}[/tex]

[tex]L(\alpha u_1 + \beta u_2) = L_2 (L_1 (\alpha u_1 + \beta u_2) )}[/tex]

[tex]L(\alpha u_1 + \beta u_2) = L_2 (\alpha L_1(u_1) + \beta L_1 (u_2) )}[/tex]

[tex]L(\alpha u_1 + \beta u_2) = \alpha L_2 (L_1(u_1)) + \beta L_2 (L_1(u_2)) [/tex]

[tex]\alpha L(u_1)+ \beta L(u_2) = \alpha L_2 (L_1(u_1)) + \beta L_2 (L_1(u_2)) [/tex]
Is this on the right track? If so, should I break it up into two pieces, and show that L1(u) is surely a mapping into V, and then show that L2(v) is surely a mapping into W?

Or am I way off
 
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  • #5
That's exactly right. Now all you have to do is note that
[tex]L_2(L_1(u_1))= L(u_1)[/tex]
and
[tex]L_2(L_1(u_2))= L(u_2)[/tex]
 

Related to Linear algebra: transformations

1. What is linear algebra and how is it related to transformations?

Linear algebra is a branch of mathematics that deals with vector spaces and linear transformations. It is closely related to transformations because linear transformations are functions that map one vector space to another, and they are fundamental to understanding linear algebra concepts.

2. What are some common examples of linear transformations?

Some common examples of linear transformations include rotation, scaling, shearing, and reflection. These transformations can be applied to various objects, such as shapes, images, and graphs, to change their size, orientation, or position.

3. How are matrices used in linear algebra transformations?

Matrices are used to represent linear transformations in a more concise and efficient manner. Each column of a transformation matrix represents the image of a basis vector, and the resulting matrix can be used to map any vector in the domain to its corresponding image in the range.

4. What is the difference between a linear transformation and a non-linear transformation?

A linear transformation preserves the properties of linearity, such as scaling and addition, whereas a non-linear transformation does not. In other words, a linear transformation follows a straight path, while a non-linear transformation does not.

5. How is linear algebra used in real-world applications?

Linear algebra and transformations are used in various fields, such as computer graphics, data analysis, physics, and economics. They are used to model and solve problems involving systems of equations, optimization, and data manipulation.

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