Representing a transformation with a matrix

[MickeyBlue]

Homework Statement

1. Use matrix multiplication to find the 2×2 matrix P which represents projection onto the line y =√3x.
2. Can you suggest another way of finding this matrix?
3. Which vectors x∈R2 satisfy the equation Px = x?
4. For which x is Px = 0?

Homework Equations

Dot product of vectors

The Attempt at a Solution

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1. I used matrix multiplication to get
(1 -√3) * ¼
(√3 3), which I know is correct

2. You can also determine this graphically, by following the same process and plotting it on a Cartesian plane.

3. This is where I'm stuck. I think my problem is notation, because is set up an equation:

(1 -√3) * ¼ °
(√3 3)
= (x)
(y)

=(x,y)

This means that (x, y) can be represented by the matrix:

¼ * (x -√3y)
(√3x 3y) ,
just using convention. However, the model answer is
(1)
λ(√3)
I know λ = x. Can someone please lead me through going from the 2x2 matrix to the vector?

 

[PeroK]

You might like to try some latex so we can see what you're doing. If you reply to this post, you can cut and paste.

$$P = \pmatrix{a&b\\c&d} $$

 

[MickeyBlue]

You might like to try some latex so we can see what you're doing. If you reply to this post, you can cut and paste.

$$P = \pmatrix{a&b\\c&d} $$

This is where I'm stuck. I think my problem is notation, because I set up an equation:

$$P = ¼*\pmatrix{1&√3\\√3&3} $$*
$$x = \pmatrix{x&\\y&} $$

=(x,y)

This means that (x, y) can be represented by the matrix:

$$x = ¼*\pmatrix{x&-√3y\\√3x&3y} $$

just using convention. However, the model answer is
λ$$x = λ*\pmatrix{1&\\√3&} $$
I know λ = x. Can someone please lead me through going from the 2x2 matrix to the vector?

 

[PeroK]

For parts 3 and 4, the question is using the boldface vector notation: $\mathbf{x} = (x, y)$. When you say "$(x, y)$ is represented by the matrix ...", then you've lost me.

 

[PeroK]

I think I see now where you are going wrong. You should have:

$$P = \pmatrix{a&b\\c&d} \pmatrix{x \\ y} = \pmatrix{ax + cy \\ bx + dy} $$

Where the right-hand side is another vector, not a matrix.

 

[MickeyBlue]

So my mistake was treating the problem as matrix multiplication instead of vector-matrix multiplication? I think I get it. Then the vector gets multiplied by the first and second colums of P respectively. Then that gets written as two linear equations. What I'm most confused about is why the final vector is represented the way it is. I'm retracting my last statement too; λ is just a variable parameter of the base vector.

 

[PeroK]

So my mistake was treating the problem as matrix multiplication instead of vector-matrix multiplication? I think I get it. Then the vector gets multiplied by the first and second colums of P respectively. Then that gets written as two linear equations. What I'm most confused about is why the final vector is represented the way it is. I'm retracting my last statement too; λ is just a variable parameter of the base vector.

You never posted your working for part 1) so I don't know how you did that. Also, I'm not sure your answer to part 2) is what is required. There are a number of methods of finding the matrix representation of an operator. In fact, when I looked at this problem, I took the answers to 3 & 4 to be the definition of the projection operator in this case! Did you define the projection by an inner product?

In any case, if you have an operator and a vector (or a matrix and a vector) then matrix multiplication still applies in the sense of multiplying a 2x2 matrix by a 2x1 matrix (which is the matrix representation of a vector) to give another 2x1 matrix (vector).

 

[Stephen Tashi]

Can someone please lead me through going from the 2x2 matrix to the vector?

Have you ever done the type of problem where you have an underdetermined systems of equations and the solution is given as a set of values involving an arbitrary parameter? One way to work such a problem is do manipulations of the coefficient matrix of the equations. Another way to work it is to use the equations themselves and the elementary algebra techniques of trying to eliminate some of the unknowns. Do you course materials expect you do the problem using matrix manipulations?

For example, the simultaneous equations:
1) $x + 3y = 2$
2) $2x + 6y = 4$

are an undetermined system because equation 2) is just a multiple of equation 1). So the problem amounts to finding solutions to equation 1).

A solution set to $x + 3y = 2$ can be expressed as "any pair of numbers of the form $( x, (2-x)/3 )$"
It could also be expressed as "any pair of numbers of the form $(\lambda, (2-\lambda)/3)$".
It could also be expressed as "any pair of numbers of the form $(2-3y, y)$"
It could also be expressed as "any pair of numbers of the form $(2 - 3\lambda, \lambda)$.
Taking for granted that an ordered pair is to be denoted $(x,y)$, it could also be expressed as "$x = 2 - 3\lambda$ and $y = \lambda$".

 

[MickeyBlue]

Have you ever done the type of problem where you have an underdetermined systems of equations and the solution is given as a set of values involving an arbitrary parameter? One way to work such a problem is do manipulations of the coefficient matrix of the equations. Another way to work it is to use the equations themselves and the elementary algebra techniques of trying to eliminate some of the unknowns. Do you course materials expect you do the problem using matrix manipulations?

For example, the simultaneous equations:
1) $x + 3y = 2$
2) $2x + 6y = 4$

are an undetermined system because equation 2) is just a multiple of equation 1). So the problem amounts to finding solutions to equation 1).

A solution set to $x + 3y = 2$ can be expressed as "any pair of numbers of the form $( x, (2-x)/3 )$"
It could also be expressed as "any pair of numbers of the form $(\lambda, (2-\lambda)/3)$".
It could also be expressed as "any pair of numbers of the form $(2-3y, y)$"
It could also be expressed as "any pair of numbers of the form $(2 - 3\lambda, \lambda)$.
Taking for granted that an ordered pair is to be denoted $(x,y)$, it could also be expressed as "$x = 2 - 3\lambda$ and $y = \lambda$".

Oh, so it's just a matter of defining the second coordinate in terms of λ