Matrix of a Linear Transformation (Abstract)

In summary, the columns of a matrix representing a transformation represent the basis set of the source vector space, while the rows represent the basis set of the range vector space. The coefficients in each column correspond to the image of the respective basis vector under the transformation. This convention is commonly used in Linear Algebra textbooks, but there may be some variations or differences in mathematical physics. Ultimately, the choice of convention is arbitrary and there is no right or wrong choice.
  • #1
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I was taught that the columns of a matrix, T, representing a transformation represent the first vector space's basis set and the rows represent the basis set of the range vector space.

i.e. T(v_k) = t_1,k*w_1 +... + t_(m,k)*w_m
So v_k would be the k-th basis vector of the first space, V, and the w's are the vector basis set for W (the range space). The coeffiicients (t's) correspond to that specific column.
In other words, a transformation of a single basis (input) element is equal to a linear combination of the range's basis.
This is the convention in Linear Algebra Done Right, wikipedia, and every text I've read... except recently on on mathematical physics, which has the reverse style (rows act like columns, colms. like rows -- as defined above). Is there a common convention? Or is one of the authors just plain wrong?
 
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  • #2
the meaning of the entries in a matrix are pure an arbitrary convention - there is no right or wrong choice. However the most common convention is this: the first column of the matrix for a linear map T represents the coefficients of the image T(e1), of the first basis vector of the source, under the map T, expanded in terms of the basis of the target.

e.g. if T maps R^2 to R^3, and e1= (1,0) and if T(1,0) = (3,4,5), then the first column will have entries 3,4,5. the second column will be the coefficients of T(e2), etc...
 

1. What is a matrix of a linear transformation?

A matrix of a linear transformation is a representation of a linear transformation between two vector spaces in terms of a rectangular array of numbers. It is used to describe how the transformation affects the coordinates of vectors in the input space to produce the coordinates of vectors in the output space.

2. How is a matrix of a linear transformation calculated?

The matrix of a linear transformation can be calculated by applying the transformation to a set of basis vectors in the input space and then expressing the resulting vectors in terms of the basis vectors in the output space. The coefficients of the resulting vectors form the columns of the matrix.

3. What is the purpose of a matrix of a linear transformation?

The purpose of a matrix of a linear transformation is to provide a concise and efficient representation of a linear transformation. It allows for easier computation and analysis of the transformation, as well as the ability to apply the transformation to any vector in the input space.

4. Can a matrix of a linear transformation be used to represent non-linear transformations?

No, a matrix of a linear transformation can only represent linear transformations between vector spaces. Non-linear transformations require more complex mathematical techniques, such as using tensors or Taylor series, to represent them.

5. How is the inverse of a matrix of a linear transformation calculated?

The inverse of a matrix of a linear transformation can be calculated by using the inverse of the basis change matrix. This is done by finding the inverse of the matrix that represents the basis change between the input and output spaces, and then multiplying it by the original matrix.

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