# Find the standard matrix of the linear transformation

• Math100
In summary, the problem involves finding the final location of a point after a transformation that includes a counterclockwise rotation by 45 degrees and a projection onto the line y=-2x. The first step can be solved using the standard matrix for rotation, while the second step involves finding the projection onto a line using orthonormal bases. The final location can be determined by applying the transformation to the original point.

None.

## The Attempt at a Solution

I know that the standard matrix of a counterclockwise rotation by 45 degrees is:
[cos 45 -sin 45]
[sin 45 cos 45]
=[sqrt(2)/2 -sqrt(2)/2]
[sqrt(2)/2 sqrt(2)/2]
But the problem says "followed by a projection onto the line y=-2x", how do I find that?

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Math100 said:

## Homework Statement

View attachment 224780

None.

## The Attempt at a Solution

I know that the standard matrix of a counterclockwise rotation by 45 degrees is:
[cos 45 -sin 45]
[sin 45 cos 45]
=[sqrt(2)/2 -sqrt(2)/2]
[sqrt(2)/2 sqrt(2)/2]
But the problem says "followed by a projection onto the line y=-2x", how do I find that?

Where does the point ##(x,y) = (a,b)## end up after the transformation?

do you know about orthonormal bases?, i.e. a basis of length one vectors and mutually perpendicular? for such a basis, it is easy to expand a vector as a linear combination of the basis vectors, using dot products. And the point is that the component in the direction of say the first vector, will be the orthogonal projection onto the line spanned by that vector. to be precise, if u is any unit length vector, and v is any other vector, then |u.v| is the length of the projection of v onto the line spanned by u.

So how do I find the point end up after the transformation?

Math100 said:
So how do I find the point end up after the transformation?

You work it out!

Where does ##(x,y) = (a,b)## go after the first step (the rotation)? So, if the rotation takes ##(a,b)## over to ##(a', b')##, where does ##(a', b')## go after the second step (the projection)?

You really are required to do the work; we are not allowed to do it for you.

## 1. What is a standard matrix of a linear transformation?

A standard matrix of a linear transformation is a matrix that represents a linear transformation in vector spaces. It is used to perform calculations and manipulations on linear transformations, such as rotations, reflections, and scaling.

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

To find the standard matrix of a linear transformation, you need to apply the transformation to the standard basis vectors. The resulting vectors will form the columns of the standard matrix. The order of the columns will correspond to the order of the basis vectors.

## 3. What is the significance of the standard matrix of a linear transformation?

The standard matrix of a linear transformation is significant because it allows for easy computation and analysis of the linear transformation. It also helps in understanding the effects of the transformation on the vectors in the vector space.

## 4. Can a linear transformation have multiple standard matrices?

Yes, a linear transformation can have multiple standard matrices. This is because the standard matrix is dependent on the choice of basis vectors. Different choices of basis vectors will result in different standard matrices for the same linear transformation.

## 5. How does the standard matrix of a linear transformation relate to matrix multiplication?

The standard matrix of a linear transformation can be used to perform matrix multiplication. This is because the standard matrix represents the transformation as a matrix, and matrix multiplication is used to apply transformations to vectors. The standard matrix can also be used to determine if a matrix is a linear transformation by checking if it satisfies the properties of a linear transformation.

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