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I think many people write their own lecture notes because they want to present the subject as it is living in their own mind, not as someone else presents it. You can really only ever try to convey your own perspective, and even in mathematics, this can be significantly different from anyone else's.

As a very small example, if you look at most linear algebra texts at how the formula for multiplying matrices is presented, you will find one of two tactics:

  1. It is just a definition, and you had better get used to it

  2. Matrix multiplication is defined the way it is so it corresponds to composition of linear functions. The proof of this is a computation which may involve a few too many summation signs for beginning students to follow fully.

My own perspective on the issue is the following:

a. I introduce matrices as record keeping devices for linear maps: the columns tell you where the basis vectors go

b. I spend time thinking about covectors, (i.e. a matrix which is just a row), and how applying a row vector to a vector is the same as taking a dot product with the transpose of the covector.

c. Realize that for a matrix M, M_{ij} = e_j^\top M e_i, since by definition Me_i is the i^{th} row, and e_j^\top of a vector just selects the j^{th} column.

d. So to find the (AB)_{ij} we just need to compute e_j^\top AB e_i = (e_j^\top A)(B e_i), which is the j^{th} row of A dotted with the i^{th} column of B. This is the standard formula, but it has been "chunked" in such a way that it makes it understandable (at least to me!).

This sequence a - d really represents thinking about a matrix as representing a bilinear form, and it is through this lens that the formula for matrix multiplication makes the most sense to me. You do not have to mention this to the students at this stage to make the sequence a-d understandable and memorable.

I find that this kind of thing occurs constantly. When I read a textbook, I usually find that I have no idea what is going on, and I have to develop some sort of narrative structure which makes sense of it. This becomes my understanding of the material. If I am teaching something, I must teach my perspective. So I often end up writing lecture notes.

p.s. If anyone knows how to format LaTeX on this site, I would appreciate it if you would let me know how.

I think many people write their own lecture notes because they want to present the subject as it is living in their own mind, not as someone else presents it. You can really only ever try to convey your own perspective, and even in mathematics, this can be significantly different from anyone else's.

As a very small example, if you look at most linear algebra texts at how the formula for multiplying matrices is presented, you will find one of two tactics:

  1. It is just a definition, and you had better get used to it

  2. Matrix multiplication is defined the way it is so it corresponds to composition of linear functions. The proof of this is a computation which may involve a few too many summation signs for beginning students to follow fully.

My own perspective on the issue is the following:

a. I introduce matrices as record keeping devices for linear maps: the columns tell you where the basis vectors go

b. I spend time thinking about covectors, (i.e. a matrix which is just a row), and how applying a row vector to a vector is the same as taking a dot product with the transpose of the covector.

c. Realize that for a matrix M, M_{ij} = e_j^\top M e_i, since by definition Me_i is the i^{th} row, and e_j^\top of a vector just selects the j^{th} column.

d. So to find the (AB)_{ij} we just need to compute e_j^\top AB e_i = (e_j^\top A)(B e_i), which is the j^{th} row of A dotted with the i^{th} column of B. This is the standard formula, but it has been "chunked" in such a way that it makes it understandable (at least to me!).

This sequence a - d really represents thinking about a matrix as representing a bilinear form, and it is through this lens that the formula for matrix multiplication makes the most sense to me. You do not have to mention this to the students at this stage to make the sequence a-d understandable and memorable.

I find that this kind of thing occurs constantly. When I read a textbook, I usually find that I have no idea what is going on, and I have to develop some sort of narrative structure which makes sense of it. This becomes my understanding of the material. If I am teaching something, I must teach my perspective. So I often end up writing lecture notes.

p.s. If anyone knows how to format LaTeX on this site, I would appreciate it if you would let me know how.

I think many people write their own lecture notes because they want to present the subject as it is living in their own mind, not as someone else presents it. You can really only ever try to convey your own perspective, and even in mathematics, this can be significantly different from anyone else's.

As a very small example, if you look at most linear algebra texts at how the formula for multiplying matrices is presented, you will find one of two tactics:

  1. It is just a definition, and you had better get used to it

  2. Matrix multiplication is defined the way it is so it corresponds to composition of linear functions. The proof of this is a computation which may involve a few too many summation signs for beginning students to follow fully.

My own perspective on the issue is the following:

a. I introduce matrices as record keeping devices for linear maps: the columns tell you where the basis vectors go

b. I spend time thinking about covectors, (i.e. a matrix which is just a row), and how applying a row vector to a vector is the same as taking a dot product with the transpose of the covector.

c. Realize that for a matrix M, M_{ij} = e_j^\top M e_i, since by definition Me_i is the i^{th} row, and e_j^\top of a vector just selects the j^{th} column.

d. So to find the (AB)_{ij} we just need to compute e_j^\top AB e_i = (e_j^\top A)(B e_i), which is the j^{th} row of A dotted with the i^{th} column of B. This is the standard formula, but it has been "chunked" in such a way that it makes it understandable (at least to me!).

This sequence a - d really represents thinking about a matrix as representing a bilinear form, and it is through this lens that the formula for matrix multiplication makes the most sense to me. You do not have to mention this to the students at this stage to make the sequence a-d understandable and memorable.

I find that this kind of thing occurs constantly. When I read a textbook, I usually find that I have no idea what is going on, and I have to develop some sort of narrative structure which makes sense of it. This becomes my understanding of the material. If I am teaching something, I must teach my perspective. So I often end up writing lecture notes.

1
source | link

I think many people write their own lecture notes because they want to present the subject as it is living in their own mind, not as someone else presents it. You can really only ever try to convey your own perspective, and even in mathematics, this can be significantly different from anyone else's.

As a very small example, if you look at most linear algebra texts at how the formula for multiplying matrices is presented, you will find one of two tactics:

  1. It is just a definition, and you had better get used to it

  2. Matrix multiplication is defined the way it is so it corresponds to composition of linear functions. The proof of this is a computation which may involve a few too many summation signs for beginning students to follow fully.

My own perspective on the issue is the following:

a. I introduce matrices as record keeping devices for linear maps: the columns tell you where the basis vectors go

b. I spend time thinking about covectors, (i.e. a matrix which is just a row), and how applying a row vector to a vector is the same as taking a dot product with the transpose of the covector.

c. Realize that for a matrix M, M_{ij} = e_j^\top M e_i, since by definition Me_i is the i^{th} row, and e_j^\top of a vector just selects the j^{th} column.

d. So to find the (AB)_{ij} we just need to compute e_j^\top AB e_i = (e_j^\top A)(B e_i), which is the j^{th} row of A dotted with the i^{th} column of B. This is the standard formula, but it has been "chunked" in such a way that it makes it understandable (at least to me!).

This sequence a - d really represents thinking about a matrix as representing a bilinear form, and it is through this lens that the formula for matrix multiplication makes the most sense to me. You do not have to mention this to the students at this stage to make the sequence a-d understandable and memorable.

I find that this kind of thing occurs constantly. When I read a textbook, I usually find that I have no idea what is going on, and I have to develop some sort of narrative structure which makes sense of it. This becomes my understanding of the material. If I am teaching something, I must teach my perspective. So I often end up writing lecture notes.

p.s. If anyone knows how to format LaTeX on this site, I would appreciate it if you would let me know how.