# Linear algebra; find the standard matrix representation

• Mdhiggenz
In summary: So In summary, the standard matrix representation for the linear operator L that reflects each vector x in R2 about the x1 axis and then rotates it 90° in the counterclockwise direction is: [0 1] [1 0]
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.

## 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?

Last edited by a moderator:
Hi Mdhiggenz!
Mdhiggenz said:
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?

I figured it out thank you!

tiny-tim said:
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.

## 1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the use of algebraic operations to solve problems related to systems of linear equations and transformations.

## 2. How is linear algebra used in science?

Linear algebra is a fundamental tool in many scientific fields, including physics, engineering, computer science, and economics. It is used to model and analyze systems, make predictions, and solve problems involving multiple variables and equations.

## 3. What is a standard matrix representation?

A standard matrix representation is a way of representing a linear transformation or system of linear equations using a matrix. It is commonly used to simplify calculations and visualize transformations in a clear and concise manner.

## 4. How do you find the standard matrix representation of a linear transformation?

To find the standard matrix representation of a linear transformation, you first need to determine the input and output dimensions of the transformation. Then, you can use the standard basis vectors for the input and output spaces to construct the matrix representation by arranging the images of the basis vectors as columns.

## 5. Can you give an example of finding the standard matrix representation?

Yes, for example, let's say we have a linear transformation T: R^2 -> R^3 that maps (x,y) to (2x+y, 3x-y, x+y). To find the standard matrix representation, we use the standard basis vectors in R^2 (i.e. (1,0) and (0,1)) and map them through T. This gives us the columns (2,3,1) and (1,-1,1), respectively. Therefore, the standard matrix representation of T is [2 1; 3 -1; 1 1].

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