How this defines a linear transformation

  • #1
158
7

Main Question or Discussion Point

Admit that [tex]V[/tex] is a linear space about [tex]\mathbb{R}[/tex] and that U and W are subspaces of V. Suppose that [tex]S: U \rightarrow Y[/tex] and [tex]T: W \rightarrow Y[/tex] are two linear transformations that satisfy the property:

[tex](\forall x \in U \cap W) S(x)=T(x)[/tex]

Define a linear transformation [tex]F: U+W \rightarrow Y[/tex] that matches with S for values in U and matches with T with values in W.

My thought is to choose the linear transformation [tex]F=S+T[/tex] because it will be the union of both transformation, right? But we have the problem that when our objects are in the intersection of both space we will get to F = 2S instead of S
 
Last edited:

Answers and Replies

  • #2
Stephen Tashi
Science Advisor
7,124
1,301
On this forum the tags using backslash and "tex" don't work for LaTex. Use a forward slash.

Admit that ##V## is a linear space about ##\mathbb{R}## and that U and W are subspaces of V. Suppose that ##S: U \rightarrow Y## and ##T: W \rightarrow Y## are two linear transformations that satisfy the property:



##(\forall x \in U \cap W)\ S(x)=T(x)##



Define a linear transformation ##F##: ## U+W \rightarrow Y## that matches with S for values in U and matches with T with values in W.
Are you asking whether it is possible to define such a linear transformaton ?

My thought is to choose the linear transformation ##F=S+T## because it will be the union of both transformation, right? But we have the problem that when our objects are in the intersection of both space we will get to F = 2S instead of S
 
  • #3
158
7
Hello Stephen!

Thanks, I edited my question.

No, I know it's possible to define the linear transformation I'm asking what should be the linear transformation and how can I get there :)
 
  • #4
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,394
180
Do you know about direct sums, and complements of subspaces?
 

Related Threads on How this defines a linear transformation

Replies
2
Views
1K
  • Last Post
Replies
7
Views
4K
  • Last Post
Replies
6
Views
8K
Replies
1
Views
2K
  • Last Post
Replies
12
Views
3K
  • Last Post
Replies
3
Views
4K
  • Last Post
Replies
2
Views
2K
  • Last Post
2
Replies
39
Views
13K
Top