Linear Combination - missing data ?

Click For Summary

Discussion Overview

The discussion revolves around a problem involving linear combinations of vectors in the context of a non-homogeneous system of equations represented by Ax=b. Participants are exploring the implications of a specific linear combination, ku-3v, being a solution to the same system.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant presents a problem where ku-3v is claimed to be a solution to the system Ax=b, questioning how to determine the value of k.
  • Another participant provides a mathematical derivation showing that A(ku-3v) leads to the equation (k-3)b = b, suggesting that since b is non-zero, k must equal 4.
  • There is a query about whether k could be 3, indicating uncertainty about the implications of the derived equation.
  • A later reply confirms that if k were 3, it would contradict the condition that b is non-zero, reinforcing that k must be 4.

Areas of Agreement / Disagreement

Participants appear to agree that k must equal 4 based on the mathematical reasoning provided, but there is an initial uncertainty regarding whether k could take other values, particularly 3.

Contextual Notes

The discussion relies on the assumption that b is non-zero, which is critical for the conclusions drawn. The implications of the linear combination and the conditions under which it holds are not fully explored.

Yankel
Messages
390
Reaction score
0
Dear all,

I am trying to solve a question, and I think that something is missing.

It is given that the vectors u and v are solutions to the non-homogeneous system of equations Ax=b.

If the vector ku-3v is a solution to the same system, then:

a) k = 4
b) k = 3
c) k = 0

The correct solution is apparently k = 4. How can you tell ?? I can't figure it out.

Thank you in advance.
 
Physics news on Phys.org
Yankel said:
Dear all,

I am trying to solve a question, and I think that something is missing.

It is given that the vectors u and v are solutions to the non-homogeneous system of equations Ax=b.

If the vector ku-3v is a solution to the same system, then:

a) k = 4
b) k = 3
c) k = 0

The correct solution is apparently k = 4. How can you tell ?? I can't figure it out.

Thank you in advance.

Hi Yankel,

You have:
$$
A(ku-3v) = kAu - 3Av = (k-3)b
$$
On the other hand, as $ku-3v$ is also a solution of the system, you have:
$$
A(ku-3v) = b
$$
Since $b\ne0$ by hypothesis, the conclusion follows.
 
So are you saying that k can't be 3 ?
 
Yankel said:
So are you saying that k can't be 3 ?
You must have $(k-3)b = b$; since $b\ne0$, this implies $k-3 = 1$ and $k=4$. This is the solution you mentioned as correct.
 

Similar threads

Undergrad Vector subspace and basis vectors in the context of data science
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Proof by induction of block diagonal decomposition of a matrix
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Characterizing linear independence in terms of span
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Solving a complex linear system with parameters
  • · Replies 11 ·
Replies
11
Views
2K
MHB Understanding Vectors: Properties and Applications
  • · Replies 4 ·
Replies
4
Views
2K
MHB Is b a linear combination of a1, a2, and a3?
  • · Replies 7 ·
Replies
7
Views
2K
MHB Combination of Linear Transformations
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad A true or false statement concerning condition number of a matrix
  • · Replies 2 ·
Replies
2
Views
1K
MHB Matrices.......whose null space consists all linear combinations
  • · Replies 3 ·
Replies
3
Views
3K