How does the row picture differ from the column picture in linear systems?

In summary, the row graph represents a system of linear equations in row form while the column graph represents a system of linear equations in column form. If there is a unique solution, the lines in the row graph intersect, and $\mathbf a_1$ and $\mathbf a_2$ are independent in the column graph. If there is no solution, either the lines in the row graph are parallel and do not coincide, or one of the equations does not have a solution at all, meaning it has no graph. If there are infinitely many solutions, we can have coinciding lines, or one of the equations has infinitely many solutions, while the other has at least 1 solution.
  • #1
otownsend
12
0
Hi,

I hope someone can help. I'm wanting to get a better grasp on the connection between the row picture v.s. the column picture of linear systems and their solutions. In the picture below, the row picture are the three graphs on the top and their corresponding column pictures are below them:

View attachment 7384

Clearly there is a pattern here in terms of what to expect the linear system of equations to look like graphically when it is represented in row form v.s. column form.

What I would like help on is the how a column picture like the one below would be represented as a row picture (sorry for it being a bit fuzzy):

View attachment 7386

I know that this solution exists and that there are infinite solutions, however I have no clue how this would be represented in the form of a column picture. I hope this makes sense.

If someone could explain this while also mentioning how this relates to linear independence and dependence that would be great.
 

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  • #2
Hi otownsend! Welcome to MHB!

In the row graph the intersection has coordinates x and y.
That corresponds to the column graph where $\mathbf b$ is a linear combination of $\mathbf a_1$ and $\mathbf a_2$ with factors x respectively y.

If we have a unique solution, the lines in the row graph intersect, and $\mathbf a_1$ and $\mathbf a_2$ are independent in the column graph, so that they can construct $\mathbf b$.

If there is no solution, either the lines in the row graph are parallel and do not coincide, or one of the equations does not have a solution at all, meaning it has no graph.
And $\mathbf a_1$ and $\mathbf a_2$ are on the same line while $\mathbf b$ is not on that line, meaning it can't be constructed, or one of $\mathbf a_1$ and $\mathbf a_2$ is zero, while $\mathbf b$ is not.

If there are infinitely many solutions, we can have coinciding lines, or one of the equations has infinitely many solutions, while the other has at least 1 solution.
I'll leave it up to you to figure out what that means for the columns graph. (Wink)
 

What is the difference between a row picture and a column picture?

The row picture and column picture are two different ways of representing a system of linear equations. In a row picture, each equation is represented as a line on a graph, and the solution is the point where all the lines intersect. In a column picture, the coefficients of the variables are represented as columns in a matrix, and the solution is found by performing row operations to reduce the matrix to its reduced row echelon form.

When should I use the row picture and when should I use the column picture?

The choice between using the row picture or the column picture depends on personal preference and the complexity of the system of linear equations. In general, the row picture is easier to visualize and can be helpful for solving simple systems of equations. The column picture is more efficient for solving larger and more complex systems of equations.

Can I convert between the row and column picture?

Yes, it is possible to convert between the row and column picture. The row picture can be converted to the column picture by setting up a matrix with the coefficients of the variables and performing Gaussian elimination to reduce the matrix to its reduced row echelon form. The column picture can be converted to the row picture by setting up equations using the columns of the matrix and solving for the variables.

Are there any limitations to using the row and column picture?

The row and column picture are only applicable for linear systems of equations, where the variables are raised to the first power. They cannot be used for non-linear systems of equations, where the variables are raised to a higher power, or for systems of equations with trigonometric or exponential functions.

Can the row and column picture be used for systems of equations with more than two variables?

Yes, the row and column picture can be used for systems of equations with any number of variables. In the row picture, each equation will be represented as a line in a graph with as many dimensions as the number of variables. In the column picture, the matrix will have as many columns as the number of variables, and the solution will be found by reducing the matrix to its reduced row echelon form.

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