Linear algebra conceptual question

In summary, the conversation discusses a linear system with a 5x3 matrix and a right-hand side vector b that is equal to a sum of two different rows of the matrix. The participants are trying to determine the number of possible solutions for this system, with one suggesting that it would result in an inconsistent system due to the mismatch in dimensions. However, they eventually conclude that the columns of the matrix represent the variables a1, a2, and a3, and that the system can have multiple solutions in the form of 5x1 matrices.
  • #1
Mdhiggenz
327
1

Homework Statement



Let A be a 5x3 matrix. If

b=a1+a2=a2+a3

then what can you conclude about the number of solutions of the linear system Ax=b? Explain

I don't have the solution for this problem, and my first thinking was the system would be overdertermined, and be most likely inconsistent. However I'm not 100% if i am even approaching the problem correctly

Thanks


Homework Equations





The Attempt at a Solution

 
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  • #2
Mdhiggenz said:

Homework Statement



Let A be a 5x3 matrix. If

b=a1+a2=a2+a3

then what can you conclude about the number of solutions of the linear system Ax=b? Explain

I don't have the solution for this problem, and my first thinking was the system would be overdertermined, and be most likely inconsistent. However I'm not 100% if i am even approaching the problem correctly

Thanks


Homework Equations





The Attempt at a Solution


What are ##a_1, a_3, a_3##? Rows of A? Columns of A? Something else?
 
  • #3
rows of a
 
  • #4
The rows of A have 3 elements each while b has 5 elements. How can ##a_1+a_2=b## possibly hold?
 
  • #5
Don't really understand
 
  • #6
Look at the dimensions of A, x, and b. A is 5x3, so what do the dimensions of x and b have to be?
 
  • #7
it would have to be 5x1?
 
  • #8
Right. Now if ##a_1## and ##a_2## represent rows of A, how many elements does each have? Can ##a_1+a_2## equal ##b##?
 
  • #9
vela said:
Right. Now if ##a_1## and ##a_2## represent rows of A, how many elements does each have? Can ##a_1+a_2## equal ##b##?

5 elements in order to equal b.

and no a1+a2 alone can not equal b. it must be coupled with 4 other rows of a1+a2. I hope I interpreted that correctly.
 
  • #10
I suspect that a1, a2, and a3 are the columns of matrix A .
 
  • #11
I think you're correct sammy I just reread the question.
 
  • #12
Mdhiggenz said:
I think you're correct sammy I just reread the question.
Can you get a solution now ?
 
  • #13
Sorry but I'm still confused, I am having a hard time transitioning to this deeper thinking math course.

But I think the answer would be that a 5x3 matrix should have an answer in the 5x1 form

So b would be could be an infinite array of 5x1 different matrices.
 

1. What is linear algebra and why is it important?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It is important because it provides the tools and techniques for solving complex problems in fields such as physics, engineering, computer science, and economics.

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