Linear Algebra: Matrix Transformation

In summary, the conversation discusses finding a matrix that represents a rotation counterclockwise around the origin by 75 degrees followed by a reflection about the x-axis. The equation for a rotation and reflection are provided, but the speaker is unsure of how to combine them. The solution is to take each transformation separately and then combine them to create a composite transformation matrix.
  • #1
Miguel Guerrero
16
0

Homework Statement


Find the matrix that represents a rotation counterclockwise around the origin by 75 degrees followed by a reflection about the x-axis

Homework Equations


I know that for A rotated counter clockwise you use the 2x2 matirx [cos(theta), -sin(theta)] [sin(theta, cos(theta)] and for a reflection about the x-axis you use the 2x2 matrix [1,0] [0,-1].

The Attempt at a Solution


To be perfectly honest, I am unsure what to do with this information. I don't want the problem solved i just need some guidance, please!
 
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  • #2
Miguel Guerrero said:

Homework Statement


Find the matrix that represents a rotation counterclockwise around the origin by 75 degrees followed by a reflection about the x-axis

Homework Equations


I know that for A rotated counter clockwise you use the 2x2 matirx [cos(theta), -sin(theta)] [sin(theta, cos(theta)] and for a reflection about the x-axis you use the 2x2 matrix [1,0] [0,-1].

The Attempt at a Solution


To be perfectly honest, I am unsure what to do with this information. I don't want the problem solved i just need some guidance, please!
Why don't you take each transformation in turn by itself? You should be able to examine your result and figure out how to make a composite transformation matrix from individual transformation matrices.
 

FAQ: Linear Algebra: Matrix Transformation

What is the definition of a matrix transformation?

A matrix transformation is a mathematical function that takes in a matrix as an input and outputs a new matrix. It involves multiplying the input matrix by a transformation matrix, which results in a new matrix with different values.

What is the purpose of using linear algebra in matrix transformations?

Linear algebra is used in matrix transformations to efficiently manipulate and analyze large amounts of data. It provides a framework for representing and solving complex systems of linear equations, which are often used in matrix transformations.

How do you know if a matrix transformation is linear?

A matrix transformation is considered linear if it satisfies the two properties of linearity: additivity and homogeneity. Additivity means that adding two matrices together before applying the transformation is the same as applying the transformation to each matrix separately and then adding the results. Homogeneity means that multiplying a matrix by a constant before applying the transformation is the same as applying the transformation to the matrix and then multiplying the result by the constant.

Can non-square matrices be used in matrix transformations?

Yes, non-square matrices can be used in matrix transformations. However, the dimensions of the input matrix and the transformation matrix must be compatible for the multiplication to be valid. For example, a 3x2 matrix can be multiplied by a 2x4 matrix, resulting in a 3x4 matrix as the output.

How are matrix transformations used in real-world applications?

Matrix transformations have many practical applications in fields such as computer graphics, data science, and engineering. For example, they can be used to rotate and scale images, analyze data sets, and solve systems of linear equations in circuit analysis or structural engineering.

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