Linear algebra; find the standard matrix representation

[Mdhiggenz]

Homework Statement

Find the standard matrix representation for each of the following linear operators:

L is the linear operator that reflects each vector x in R² about the x₁ axis and then rotates it 90° in the counterclockwise direction.

Homework Equations

The Attempt at a Solution

So my thinking is they wat a L.O that reflects any vector about the x1 axis. So for instance if we input a vector e1=(1,0) into our linear operator
L(e1)=(x1,-x2)
L(e1)=(1,0)^T

Same goes for e2

e2=(0,1)

L(e2)=(y1,-y2)

L(e2)=(0,-1)

Here is where I get a bit confused. How do It ake in the 90° counter clockwise rotation into account?

 

[tiny-tim]

Hi Mdhiggenz!

How do It ake in the 90° counter clockwise rotation into account?

You start with a general (x,y), not just (1,0) and (0,1) …

where does (x,y) go to on the reflection? and then where does that go to on the rotation?

 

[Mdhiggenz]

I figured it out thank you!

 

[HallsofIvy]

Hi Mdhiggenz!


You start with a general (x,y), not just (1,0) and (0,1) …

where does (x,y) go to on the reflection? and then where does that go to on the rotation?

I disagree. Seeing what a linear transformation does to the basis vectors is a standard way of finding a matrix representation of it: Apply the linear transformation to each domain basis vector in turn, writing the result as a linear combination of the range basis vectors. The coefficients for a column of the matrix.

The linear transformation is: "reflect each vector x in R2 about the x1 axis and then rotate it 90° in the counterclockwise direction".

Reflecting (1, 0) in the x1 axis doen't change it but rotating 90° in the counterclockwise direction gives (0, 1).

Reflecting (0, 1) in the x1 axis gives (0, -1) and rotating 90° in the counterclockwise direction gives (1, 0). That is (1, 0) is mapped to (0, 1) and (0, 1) is mapped into (1, 0). This linear transformation is the same as reflection about y= x.