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Hey all, my dual enrollment course uses Gilbert Strangs Introduction to Linear Algebra 6th Edition, however I'm kinda frustrated with the emphasis on computation rather than understanding. I went through a decent chunk of LADR but I'm finding it hard to keep track of what I'm "meant" to be learning for class, so I was wondering if anyone had a recommendation that followed the content of Strang, but had a larger emphasis on proofs.

Also, does anyone now of any good books on ODE's?

This material covers applications of linear algebra to engineering and quantum mechanics. The mathematical steps are presented in as much detail as possible so that undergraduate students can easily follow them.

Recently i started learning Linear Algebra, din`t learn on University,
Now i`m feeling useful again.
I created this application using Python, ThreeJS on PySide6

Is a plotter 3D, i`m making a for creating notebooks, like TeX with plotter, help me to understand the logics

Hey everyone! I recently made a video explaining Choi matrices, complete positivity, and quantum channels.

If you're studying linear algebra and are curious about how concepts like eigenvalues, positive semidefinite matrices, tensor products, and the trace are used in quantum computing, I thought this might be interesting!

I work through several examples step by step, including how a matrix can have a negative eigenvalue and what that tells us about whether a quantum operation is physically valid.

Thought I'd share since it's a nice real-world application of linear algebra!

Happy to hear any feedback or suggestions!

The usual definition of a vector space over the real numbers R is the tuple defined by the cartesian product R^n = R x ... x R = {(x_1,..., x_n) | x_i in R}. An equivalent, but different, definition is the set exponential R^n = R^{1,...,n} = {x: {1,...,n} -> R} where x_i corresponds to x(i).

The connection between cartesian products and set exponentials is currying: (A x B)^C is equivalent to A^(B^C). A function f: A x B -> C corresponds to a function g: A -> (B -> C) by g(a)(b) = f(a, b). A function g: A -> (B -> C) corresponds to a function f: A x B -> C by f(a, b) = g(a)(b).

Exercise: Show linear operators from R^n -> R^m can be identified with vectors R^{n x m}.

I have just began to learn Linear Algebra, and after being confused by response of DeepSeek in the matter, I came here.

Question:

If dim(V)=n, then according to some theorem (please don't take offense since i don't remember name), any spanning set of n vectors is a basis.

My question is, aren't the elements in that set linearly independent? DeepSeek said not necessarily. What does that mean?
Math StackExchange had some answers agreeing, although one "great mind" said this:

No.

For example (1,0,0) and (0,1,0) are linearly independent, but don't span R3.

For example (1) and (2) spans R1 but are not linearly independent.

Just to confirm, i am not asking about B => A. I am asking about A => B, if I am saying it right.

___

So, what is the correct answer?

i've been working on a tool to help you create books & notes with ease

and one of it features is allowing you to write mathematical Equations in Latex

it also has co-author mode which creates any latex equation in seconds with simple prompts

One useful bridge from linear algebra to quantum mechanics is to compare an operator that labels a state with operators that connect different states.

For the ideal quantum harmonic oscillator, the Hamiltonian is ħω times the sum of the number operator and one-half times the identity. Its energy eigenstates are the number states. Acting on a normalized number state, the lowering operator gives the next lower state multiplied by √n; the raising operator gives the next higher state multiplied by √(n + 1). The lowering operator sends the ground state to zero.

Those square-root factors are part of the construction, not optional decoration on an energy-level diagram. The ladder method produces the same normalized Hermite–Gaussian eigenfunctions as the differential-equation method, up to a phase for each state.

One caveat if you explore this with matrices: a finite truncation introduces an artificial top state. The full canonical commutator does not hold at that boundary, so the truncated matrix is not an exact model of the entire oscillator algebra.

I made this Blake Pi Physics animation to connect the operator picture with the differential-equation picture:

https://youtu.be/7_DcdXVh5mc

For people learning this transition, is it more helpful to see the operators act on the states first, or to start from the Hamiltonian's factorization?

This material does not focus strictly on abstract pure linear algebra; rather, it deals with special functions applied to engineering mathematics and quantum mechanics, specifically within the framework of linear differential operators and self-adjoint eigenvalue problems.
It provides a detailed, step-by-step mathematical derivation showing why the Legendre series must truncate to a finite polynomial to avoid logarithmic boundary divergence at x = ±1.

Why do some vectors when multiplied by Another vector yield a vector and others yield a scalar. I'm aware of dot and cross products. But what's the physical significance of 'multiplying' vectors? I mean... I know what it means to actually "multiply" scalars. But.... Is vector multiplication just a name which can either mean projection of a vector onto another and the area or volume of 2 corresponding vectors?

Or is multiplication of vectors just a naming thing which doesn't have ANYTHING to do with how we understand scalar multiplication?

Expanding while being taught this term

https://people.maths.bris.ac.uk/~ki18754/LA_notes/LA_notes.html

Hi everyone,

I’m a college student currently taking my first Linear Algebra course. My ultimate career goal is to go into robotics, so I know this course is critical and that this semester is just the beginning of a long journey with this math.

My brain has always been wired to apply—simply applying formulas, calculating matrices, and finding numbers. However, this course introduces a lot of proofs and abstract concepts, and I’m having a hard time wrapping my head around them. I feel like I'm stuck memorizing procedures rather than genuinely understanding why things work geometrically or structurally.

I really want to upgrade my thinking and install a proof/conceptual understanding in my brain. Since my time is limited this semester, I know I won't master everything, but I want to build a solid foundation.

- How did you train your brain to understand proofs?

- Are there specific intuition-building resources, mental models, or study strategies you recommend for someone heading into robotics? (I’ve heard of 3Blue1Brown, but I'd love to know how to bridge the gap between his visual intuition and rigorous proof writing.)

Thank you so much for any advice!

I’m currently studying eigenvalues in linear algebra, and I understand the definitions of algebraic and geometric multiplicity, but I’m still struggling to understand the actual difference between them.

If someone could explain it in a simple, intuitive way, preferably with a small example, I’d really appreciate it.

Struggling with graph transformations. Like all of it.

This material shows a real application of linear algebra.
I showed all the math steps without skipping any equations, so that 1st and 2nd-year students can easily follow.
I hope this helps you!

did I solve it right?/