How is equation transformed by matrix

[montana111]

Homework Statement

Consider in R^2 the points satisfying the equation 2x_2 − x_1 − 2 = 0. Show on a plot
the points satisfying this equation. How are these points transformed by the matrix
A =
[
1 −1
−1 2
]

Hint: To draw the plot, use the fact that two points determine a line.

Homework Equations

The Attempt at a Solution

what does the question mean when it asks "how is the equation transformed by the matrix"? I don't really understand what they are asking me to do.

Thanks!

 

[lanedance]

2x_2 − x_1 − 2 = 0 is the equation of a line

now take any point on that line x=(x1,x2)^T and consider the product Ax, where is that point mapped too,

you could consider it for the whole line (ie every point). What is the image of teh line when multiplied by A? is it another line?

 

[montana111]

2x_2 − x_1 − 2 = 0 is the equation of a line

now take any point on that line x=(x1,x2)^T and consider the product Ax, where is that point mapped too,

you could consider it for the whole line (ie every point). What is the image of teh line when multiplied by A? is it another line?

1) what do you mean by raising the point (x,y)^T ?

2) I still don't understand what I am supposed to do with regards to the equation and the matrix A. It sounds like maybe i find a point on the line in the equation (lets say (29, 8) which prob isn't on the line but just for example) and then multiply it by A?

 

[Hammie]

One way to try this problem would be to parametrize your line, and express it as a vector, then see how your matrix will map it.

 

[Hammie]

Or you could use the hint. You have an equation of a line. Find the coordinates for two points. If you have two coordinates, can you find the vector?

Matrices transform vectors to vectors by multiplication.

 

[montana111]

So i got some points on that line. I used (2,2) and (0,1). Then i drew the plot. then i multiplied A by (2, 2). Multiplying A by (2, 2) seems incorrect. Here is what it looks like anyways...

A = [ <1; -1> , <-1; 2> ] [<2; 2>] = [ <2; -2> , <-2; 2> ] (where <stuff> is meant to be a column)


I feel like I am doing something wrong. Any ideas?

 

[HallsofIvy]

A straight line is determined by two points. Since a linear transformation (multiplication by a matrix) will map a line to a line, pick two points on the given line and multiply each by the matrix to get two points that determine the new line.

 

[lanedance]

So i got some points on that line. I used (2,2) and (0,1). Then i drew the plot. then i multiplied A by (2, 2). Multiplying A by (2, 2) seems incorrect. Here is what it looks like anyways...

A = [ <1; -1> , <-1; 2> ] [<2; 2>] = [ <2; -2> , <-2; 2> ] (where <stuff> is meant to be a column)


I feel like I am doing something wrong. Any ideas?

the first entry should be 1.2+(-1).2=0