Linear algebra conceptual question

[Mdhiggenz]

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

 

[Ray Vickson]

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?

 

[Mdhiggenz]

rows of a

 

[vela]

The rows of A have 3 elements each while b has 5 elements. How can $a_1+a_2=b$ possibly hold?

 

[Mdhiggenz]

Don't really understand

 

[vela]

Look at the dimensions of A, x, and b. A is 5x3, so what do the dimensions of x and b have to be?

 

[Mdhiggenz]

it would have to be 5x1?

 

[vela]

Right. Now if $a_1$ and $a_2$ represent rows of A, how many elements does each have? Can $a_1+a_2$ equal $b$?

 

[Mdhiggenz]

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.

 

[SammyS]

I suspect that a₁, a₂, and a₃ are the columns of matrix A .

 

[Mdhiggenz]

I think you're correct sammy I just reread the question.

 

[SammyS]

I think you're correct sammy I just reread the question.

Can you get a solution now ?

 

[Mdhiggenz]

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.