Guidance on Matrices: Get a Better Understanding with Books/Videos

[Yashbhatt]

Hello, I have been studying matrices and determinants recently and do not understand why certain things are done the way they are. Like, why is matrix multiplication defined the way it is.

I find that there are not enough proofs. Is there any book/article/video that any of you recommend to gain a better understanding of matrices and determinants?

 

[micromass]

There is a bit of a pedagogical problem when introducing matrices and determinants. They are very useful tools and there are very good reasons to define them the way we do. But it is typically not immediately possible to tell the students about these reasons.

For multiplication of matrices, one preliminary reason could be that it gives us an easier notation to denote systems of linear equations. But the real reason of course is that multiplication of matrices come from a very geometrical object, namely the linear maps. Once you understand linear maps, you'll be convinced of the necessity of the matrix multiplication.

The determinant is a very complicated expression. But it essentially has one primary use, namely that it decides how many solutions a system of equations have. Or equivalently, it decides whether a matrix is invertible or not. There are many geometrical ways of seeing the determinant, a favorite one is that the determinant gives the volume of a parallelepiped.

A good introductory book (with proofs) is "Introduction to Linear Algebra" by Lang (be sure not to get his more advanced "Linear algebra"). https://www.amazon.com/dp/0387962050/?tag=pfamazon01-20
A very very good book (but also more advanced) is Treil's "Linear algebra done wrong": http://www.math.brown.edu/~treil/papers/LADW/LADW.html It is completely free too.

 

[Yashbhatt]

There is a bit of a pedagogical problem when introducing matrices and determinants. They are very useful tools and there are very good reasons to define them the way we do. But it is typically not immediately possible to tell the students about these reasons.

For multiplication of matrices, one preliminary reason could be that it gives us an easier notation to denote systems of linear equations. But the real reason of course is that multiplication of matrices come from a very geometrical object, namely the linear maps. Once you understand linear maps, you'll be convinced of the necessity of the matrix multiplication.

The determinant is a very complicated expression. But it essentially has one primary use, namely that it decides how many solutions a system of equations have. Or equivalently, it decides whether a matrix is invertible or not. There are many geometrical ways of seeing the determinant, a favorite one is that the determinant gives the volume of a parallelepiped.

A good introductory book (with proofs) is "Introduction to Linear Algebra" by Lang (be sure not to get his more advanced "Linear algebra"). https://www.amazon.com/dp/0387962050/?tag=pfamazon01-20
A very very good book (but also more advanced) is Treil's "Linear algebra done wrong": http://www.math.brown.edu/~treil/papers/LADW/LADW.html It is completely free too.

Thanks. I'll try them. How advanced are linear maps?

 

[micromass]

Thanks. I'll try them. How advanced are linear maps?

Depends on which book you read. But most introductory books treat them rather well. They should be easy once you realize that they are nothing more than generalization of very familiar things: rotations, reflections, etc.

 

[QuantumCurt]

The determinant is a very complicated expression. But it essentially has one primary use, namely that it decides how many solutions a system of equations have. Or equivalently, it decides whether a matrix is invertible or not. There are many geometrical ways of seeing the determinant, a favorite one is that the determinant gives the volume of a parallelepiped.

I don't think I really understood determinants qualitatively until I used the triple scalar product in Calc 3 to find the volume of a parallelepiped.

A good introductory book (with proofs) is "Introduction to Linear Algebra" by Lang (be sure not to get his more advanced "Linear algebra"). https://www.amazon.com/dp/0387962050/?tag=pfamazon01-20
A very very good book (but also more advanced) is Treil's "Linear algebra done wrong": http://www.math.brown.edu/~treil/papers/LADW/LADW.html It is completely free too.

I second the suggestion of Lang's book. As you point out, be sure to get the intro book. I accidentally got 'Linear Algebra' instead of 'Intro to Linear Algebra' at first, and quickly got in very well over my head.

 

[Akhand]

Hello, I have been studying matrices and determinants recently and do not understand why certain things are done the way they are. Like, why is matrix multiplication defined the way it is.

I find that there are not enough proofs. Is there any book/article/video that any of you recommend to gain a better understanding of matrices and determinants?

give enough time to the problems. do not see your watch again and again. plan for atleast 1 hour a day for matrices. then after somedays interest will be build up