Linear transformation and matrix transformation

In summary, not all linear transformations are matrix transformations, but every linear transformation can be represented by a matrix multiplication. This requires selecting a specific basis, and if using the standard basis for R^n to R^m, then every linear transformation can be identified with a specific matrix and vice-versa. However, there are examples of linear transformations that are not matrix transformations, such as those involving non-matrix vector spaces and mappings from a subspace of R-3 to R-2. These can still be represented by matrices if a coordinate mapping is used, but this is not the case for all vector spaces. David Lay's textbook has some confusing claims about this topic, but it becomes clear later on in the text. Additionally, there are
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Ali Asadullah
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Do all linear transformations are matrix transformation? In a book by David C Lay, he wrote on page 77 that not all linear tranformations are matrix transformations and on page 82 he wrote that very linear transformation from Rn to Rm is actually a matrix transformation. I know that every matrix transformation is linear but not sure about the reverse.
 
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Every linear transformation can be represented by a matrix multiplication. But writing a linear transformation as a matrix requires selecting a specific basis. If you are talking about [itex]R^n[/itex] to [itex]R^m[/itex] (there are other vector spaces) and are using the "standard" basis, then, yes, you can identify any linear transformation with a specific matrix and vice-versa.
 
  • #3
See this post for more about the connection between linear operators and matrices.
 
  • #4
Ali Asadullah said:
Do all linear transformations are matrix transformation? In a book by David C Lay, he wrote on page 77 that not all linear tranformations are matrix transformations and on page 82 he wrote that very linear transformation from Rn to Rm is actually a matrix transformation. I know that every matrix transformation is linear but not sure about the reverse.

I was wondering the same when I read that bit in the textbook! To answer your question, examples of linear transformations that are not matrix transformations are those that involve non-matrix vector spaces (eg. the vector space of polynomials) and the mapping from a planar subspace of R-3 onto R-2. These examples are given later in the text too, but unfortunately David Lay does not take the trouble to point out that these linear transformations in themselves are not matrix transformations (relating them back to his earlier claim that you quote). Note however that every vector space can be coordinate-mapped onto R-n, giving each of their vectors a unique column vector representation (for example, the coordinate vector of 4+ 3t^2 relative to the standard basis for P-2 is (4,0,3)). Thus, every linear transformation from a vector space V has a unique matrix representation after all! (The matrix acts on the coordinate vectors of the vectors in V, not the vectors in V themselves.)

David Lay's textbook is horrible as a reference text because the material is all over the place (especially on linear transformations), but he has some valid pedagogical reasons for structuring the book the way he does. He shouldn't have made that claim though, as it's an unimportant technicality at that point in the text that causes unnecessary confusion yet becomes patently self-evident later on.

Hope this message helps any future readers of David Lay's text!

ps. Btw, do think about _why_ a linear transformation cannot be a matrix transformation when the domain/codomain is a proper subspace of R-n (i.e. does not span the whole of R-n).:smile:

pps. For completeness, let me state that each matrix can of course represent many different linear transformations.
 
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There are infinite dimensional vector spaces. Is the definition of linear transformation being discussed in this thread restricted to a mapping from one finite dimensional vector space to another? (I assume the definition of "matrix" that is being discussed refers to a finite dimensional array.)
 
  • #6
Certainly, when I said "any linear transformation can be represented as a matrix", I was thinking of the finite dimensional case. Thanks for clarifying that.
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a linear manner. This means that the output is a linear combination of the input vectors, and the transformation preserves the properties of vector addition and scalar multiplication.

2. How is a linear transformation represented?

A linear transformation can be represented by a matrix. Each column of the matrix represents the image of a basis vector from the input space, and the transformation is determined by multiplying the matrix with the input vector.

3. What is the difference between linear and non-linear transformations?

A linear transformation preserves the properties of vector addition and scalar multiplication, while a non-linear transformation does not. In other words, the output of a linear transformation is a straight line or a plane, while the output of a non-linear transformation can be curved or distorted.

4. How is a matrix transformation different from a linear transformation?

A matrix transformation is a type of linear transformation where the transformation is defined by a matrix. However, not all linear transformations can be represented by a matrix, as the transformation must be a linear combination of the input vectors.

5. What are some real-world applications of linear transformations?

Linear transformations have various applications in fields such as computer graphics, image processing, and data analysis. For example, linear transformations are used to rotate, scale, and translate images in computer graphics, and to reduce the dimensionality of data in machine learning algorithms.

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