How Do You Apply Linear Transformations to Find T(-3, 4)?

Click For Summary

Homework Help Overview

The problem involves applying a linear transformation T to the vector (-3, 4) in the context of linear algebra. The original poster presents a linear transformation defined by its action on two basis vectors, u and v, and seeks to determine T(-3, 4) by expressing the vector as a linear combination of u and v.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the necessity of expressing (-3, 4) as a linear combination of the basis vectors u and v, with some exploring the implications of linear independence and the coefficients involved in the combination.

Discussion Status

Some participants have provided guidance on how to express the vector as a linear combination, and there is a confirmation of the correctness of the computed coefficients. However, the discussion does not reach a consensus on the final outcome of T(-3, 4) as it is presented in a manner that suggests further exploration may be warranted.

Contextual Notes

The original poster's approach relies on the hint provided to express the vector in terms of the basis vectors, which may imply constraints on the methods available for solving the problem.

Precursor
Messages
219
Reaction score
0

Homework Statement


Let u = (1,2), v = (3,1) and T: [tex]R^{2}\rightarrow R[/tex] be a linear transformation such that T(u)= 4 and T(v)= 5. What is T(-3, 4)? (Hint: Write (-3,4) as a linear combination of u and v.)


Homework Equations


T(x)= b


The Attempt at a Solution


I don't really know where to start. I know I have to write (-3,4) as a linear combination of u and v, but what do I do from there?
 
Physics news on Phys.org
The vectors u and v form a basis for R^2, since they're linearly independent (and there is two of them). So, you can write any vector in R^2 as their linear combination, with some coefficients α, β, so, for xome vector x (in your case (-3, 4) ), you have x = α u + β v. Now, what does T(x) equal to?
 
So I determined α = 3 and β = -2 using your method. I was then able to solve for T(-3,4) = 2.

Is this correct?
 
Yes, it is correct.
 

Similar threads

  • · Replies 26 ·
Replies
26
Views
3K
  • · Replies 1 ·
Replies
1
Views
3K
  • · Replies 7 ·
Replies
7
Views
1K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 2 ·
Replies
2
Views
1K
Replies
1
Views
2K
  • · Replies 1 ·
Replies
1
Views
1K
Replies
12
Views
2K
  • · Replies 24 ·
Replies
24
Views
4K
  • · Replies 18 ·
Replies
18
Views
3K