Matrix Representation of a linear operator

In summary, T is a linear operator that operates on 2 by 2 matrices over the complex plane and outputs the trace of the matrix. It can be shown that T is linear and can be represented as a 1 by 4 matrix with respect to a basis consisting of standard basis vectors for 2 by 2 matrices. This matrix should be multiplied by a 4 by 1 column vector representation of the vector in Rn, rather than a 2 by 2 matrix.
  • #1
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T is a linear operator from the space of 2 by 2 matrices over the complex plane to the complex plane, that is
T: mat(2x2,C)[tex]\rightarrow[/tex]C, given by

T[a b; c d] = a + d

T operates on a 2 by 2 matrix with elements a, b, c, d, in case that isn't entirely clear. So T gives the trace of the matrix, and it can be shown that T is linear.

I'm having trouble finding a matrix representation of T with respect to any basis.
If T can be written as a matrix with respect to a basis, then that matrix applied to the 2 by 2 above would give a 1 by 1 matrix. I don't understand how this can be the case, since you can't multiply a 2 by 2 matrix with any matrix that I know of to get a 1 by 1 matrix as the result.

If I go by the rule that the columns of the matrix are the images of T applied to the basis which you are finding the matrix representation with respect to, then using the basis vectors [1 0; 0 0], [0 1; 0 0], [0 0; 1 0], [0 0; 0 1], which I think are the standard basis vectors for 2 by 2 matrices, I end up with a 1 by 4 matrix of 0s. This doesn't seem right, because I can't multiply a 1 by 4 matrix with a 2 by 2 matrix to get a 1 by 1 matrix.

Can someone please explain to me where I'm going wrong with this?
Thanks
 
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  • #2
The matrix will be 1×4, and you won't be multiplying it with any 2×2 matrices, but with a 4×1 matrix.

This should be easy to see if you understand the relationship between linear operators and matrices described in this post.
 
  • #3
When you represent a linear operator as a matrix, you don't multiply the matrix by the vector, you multiply the matrix by the representation of the vector in Rn.
That is, you can represent the matrix
[tex]\begin{bmatrix}a & b\\ c & d\end{bmatrix}[/tex]

as the column
[tex]\begin{bmatrix}a \\ b \\ c\\ d\end{bmatrix}[/tex]
 
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1. What is a matrix representation of a linear operator?

A matrix representation of a linear operator is a way of representing a linear transformation from one vector space to another using a matrix. It allows us to perform operations on vectors and matrices in a concise and efficient manner.

2. How is a matrix representation of a linear operator calculated?

To calculate the matrix representation of a linear operator, we first need to choose a basis for the vector space. Then, we apply the linear operator to each basis vector and write the resulting vectors as columns of a matrix. The resulting matrix is the matrix representation of the linear operator.

3. What are the advantages of using a matrix representation of a linear operator?

One advantage of using a matrix representation of a linear operator is that it simplifies the computation of linear transformations. Instead of performing the transformation on each vector individually, we can simply multiply the matrix representation by the vector to get the transformed vector. This is particularly useful when dealing with large matrices or performing multiple transformations.

4. Can a matrix representation of a linear operator change?

Yes, the matrix representation of a linear operator can change depending on the choice of basis for the vector space. Different bases will result in different matrix representations, but they will all represent the same linear operator.

5. How is a matrix representation of a linear operator used in applications?

A matrix representation of a linear operator has many applications in mathematics, physics, and engineering. It is commonly used in solving systems of linear equations, calculating eigenvalues and eigenvectors, and in computer graphics and image processing. It also plays a crucial role in the study of linear algebra and its applications.

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