I'd recommend watching these videos for some motivation: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
And remember, linear algebra has broader application than just geometry!
Fundamentals of Linear Algebra and Optimization [pdf]
21–30 of 63 posts
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#22My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
To this book's credit, it covers a LOT more material than Axler's book. And it's more heavily focused on optimization methods for which the details of tabular representations of matrices takes center stage.
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#23My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
Strang’s textbook (and lectures) introduce matrix multiplication AB as combinations of A’s columns. He uses the cell-by-cell / “dot product” formulas just to check his work, which seems reasonable to me. I’m a big fan of “combinations of columns”. I sometimes even hear that in Strang’s voice when I’m doing some matrix multiplications.
Axler was way too abstract for me as a first linear algebra book. I look forward to returning to it once I finish Strang.
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#24My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
I'm curious what you think of the "No Bullshit Guide to Linear Algebra" [1]? I'm considering buying it to refresh my knowledge from school. Or what books do you suggest?
[1] https://www.amazon.com/No-bullshit-guide-linear-algebra/dp/0...
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#25My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
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Re: Fundamentals of Linear Algebra and Optimization [pdf]
#26My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
IIRC, Apostol's Calculus, volume 2, does it right (linear transformations first, then matrices).
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#27Did fields other than R and C eat the authors' children or something?
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#28My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
Starting with an abstract formal description of vector spaces and linear transformations is not a pedagogically useful introduction, and it leaves out a huge amount of the historical/motivational context explaining most of the conventions used in linear algebra, and even the mental/conceptual understanding most working mathematicians have about the meaning of linear models and their use in various parts of mathematics (not even to mention people in other STEM fields).
The two places to start historically (and easily accessible to high school students) are:
(1) understanding and working with displacement vectors in 3-dimensional Euclidean affine space and in general thinking about transformation geometry (sometime later this can be extended to other kinds of non-Euclidean or non-metrical geometry), especially with reference to problems in Newtonian mechanics.
(2) systems of linear equations: this one is inherently coordinate-heavy and matrix based, at least to start out, and explains our conventions for how matrices are written, index order, multiplication of matrices by "column" vectors on the right, the use of matrix equations to fold several equals signs into one, etc.
After that I'd call out the manipulation of vectors of polynomial coefficients as an accessible additional concrete example of a linear space.
Discussion of other linear spaces or more purely abstract treatments proving properties from axioms can come sometime later, after students have familiarity with some of those tools, and after they have applied them to some problems in statistics, multivariable calculus, computer graphics, ODEs, optimization, etc.
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#29My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
Do you have any other tests for calculus, analysis, abstract algebra, discrete mathematics, probability or statistics books?
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#30My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
> My greatest wish in STEM education is that we teach linear algebra better.
That's easy. Vector spaces and linear transformations are best understood abstractly. Matrices are best understood as visually and computationally convenient representations of tensor products, so don't mention matrices at all until well after you have established the basic properties of tensor products, in particular, the natural isomorphism between `L(V,W)` and `Dual(V) (x) W`.