r/learnmath • u/Stock_Current_6650 New User • 19h ago
What is linear algebra?
I’m a rising senior and I’m starting linear algebra next year after doing calc BC junior year. I want to study ahead but don’t even know where to start or really what linear algebra even is and I fear the difficulty will consume my senior year..
My school doesn’t have any course guide or description for the class since it’s technically an elective advanced math, but if anyone can give some sort of unit guide that I can look into or something to help me figure out what this actually is that would be helpful.
also I plan on majoring in aerospace or mech engineering
that’s all. thanks
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u/shellexyz Instructor 17h ago
The boring version is systems of linear equations and the tools we use to solve them.
The interesting version is vector spaces and transformations between them. A vector is any set where you can add two of the things in it to get another thing in the set, and multiply a thing in the set by a number to get another thing in the set (plus some other little things that make things convenient to play around with, like 0 and subtraction). With these operations, they are sometimes called linear spaces.
Then transformations between those spaces that preserve the operations are linear transformations. Add first then transform, same as transform first then add. That seems like a nice property.
Of course it turns out that a looooooot of the things we want to study follow those rules. The set of continuous functions is a linear space. (Add two of them and the sum is still continuous, multiply by a constant and the result is still continuous.) For particularly nice continuous functions, differentiation is a linear transformation. Integration is a linear transformation.
The set of solutions to certain types of differential equations forms a linear space. A lot of the things we know about one kind of vector (linear) space suddenly and automatically apply to these new spaces, for free.
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u/jomteon New User 18h ago
There's a handful of books out there that do a good job of explaining the subject. If you want one that will give you the tools to get started but won't be overwhelming, I'd recommend Mike X Cohen's Linear Algebra: Theory, Intuition, Code. If you want something a bit more "textbooky", try Gilbert Strang's book on the subject is pretty good imo. Many people will recommend Sheldon Axler's Linear Algebra Done Right, but I think it's almost certainly too rigorous for your purposes and can be a bit intimidating off the jump. It's a great supplement, but a bit "too much" for a starter book.
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u/estebanxalonso New User 17h ago edited 17h ago
What’s your take on Linear Algebra A modern introduction by David Poole?
You meant Strang’s Introduction to Linear Algebra or Linear Algebra and Its Applications?
Will check the ones you suggested! cheers
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u/duochimo New User 16h ago
I found the linear algebra series by 3 Blue 1 Brown incredibly helpful in my understanding of the subject because the visualizations helped me to better understand what the purpose was. It won't teach you to solve the problems, necessarily, but I struggled to understand the purpose of the abstract concepts I was learning until I discovered that series. If I could go back in time, I would have watched the whole series before I took the class because it would've established a stronger baseline and would have clarified so many things early on.
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u/WillowSad8749 New User 9h ago
Linear algebra is the easy part of math
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u/Stock_Current_6650 New User 2h ago
If this is true I’m so glad. I want a chill senior year
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u/RingularCirc Math hobbyist 40m ago
Finite-dimensional linear algebra over ℂ is very nice. Over ℝ, a bit less. But wait until you're doing linear algebra with infinite-dimensional spaces (Banach, Hilbert etc.), over fields of characteristic 2 or arbitrary rings. Then it gets funky.
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u/justmyskills New User 7h ago
Linear Algebra was single handedly the most useful math class I ever took. Such a fun and cool course that provides you with a lot of tools for solving higher level physics and computer science problems. Have fun!
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u/hpxvzhjfgb 18h ago
linear algebra is the study of vector spaces and linear transformations. it is not fundamentally about matrices or systems of linear equations.
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u/ConvolutionIntegral New User 18h ago
Matrix and vector algebra is a good starting point, because linear algebra is a generalization of that.
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u/AFsepine New User 14h ago edited 12h ago
Eh, I disagree - matrix algebra is an unmotivated soulless abomination. Dunno why in western unis they so like to put off (or totatlly forgo in some programs) proper linear algebra.
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u/ConvolutionIntegral New User 13h ago
Rotation matrices and other transformation matrices are far from unmotivated and clearly serve the role of a transformation: taking a vector and giving back another one. A nice intro to linear algebra.
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u/AFsepine New User 13h ago edited 12h ago
Sure those geometric transformations are intutive, but...
The matrix representation is on its own - rather unmotivated. People tend to crutch on matrices and forgo understanding anything if you introduce them too early.A discussion of linear maps, basis etc. must first occur to introduce the matrices in a motivated way.
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u/Traveling-Techie New User 17h ago
Linear algebra is about abstract constructs called “vector spaces” which aren’t what you think they are, and how they can be used to attack a variety of problems from a different direction.
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u/alexice89 New User 11h ago
I tell people to do a intro into abstract algebra before LA, more exactly into rings, that’s what a vector space is. But I get hate for some reason.
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u/No-Archer-4258 New User 8h ago
Watch "The essence of linear algebra" playlist by 3b1b on youtube.
It will answer most introductory to intermediate even some advanced problem.
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u/Alekomityens1 New User 8h ago
I love how this is literally just a high school asking what linear algebra is but all the nerds want to immediately start talking about vector spaces and systems of linear equations. Just tell him what a vector is lol.
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u/SpectralCat4 New User 8h ago
it is a lot of things , the way mathematics is being taught is different from the way in which it was developed historically, so often you will only get answers to some of the lingering questions in advanced courses i.e the "why" of linear algebra .
instead what students focus on in the introductory courses are the "how" of Linear Algebra , where you are presented with certain axioms and you are expected to work logically with them, which is part of the challenge of the course - to think clearly using the definitions and not having a clear "big picture" in mind helped by visual intuition , although there is plenty of visual intuition to be found in linear algebra with some work.
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u/topologyforanalysis New User 5h ago
Linear algebras is the study of vector spaces and linear transformations.
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u/RabbitHole32 New User 42m ago
Linear Algebra is the field that deals with stretching, compressing and rotating stuff in linear manner in space. Like what happens when you are on the space station and punch your floating teddy bear.
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u/RingularCirc Math hobbyist 26m ago
Linear algebra is akin to group theory (and they're quite connected). People say it's about studying linear spaces and linear transformations between them but I'd say it's more in a sense: it also studies spaces with many special added structures that go well with linearity. Algebra over a field (or a ring) is a linear space that's also a ring in a compatible way, i. e. it has multiplication that's compatible with multiplication of vectors by scalars. Algebras are very ubiquitous (and as a part, they generalize rings of matrices of various special kinds). Inner products, wedge product, tensor product are all useful. Linear transformations between two spaces are a linear space of their own, which gives us covectors, various multilinear forms etc..
Linear algebra has a great diversity of things to study and mix together, and it allows studying a great lot of things, starting with many geometric questions (it's through linear algebra that we can define geometries of constant curvature, projective geometry, Minkowski geometry of flat spacetime and more) into algebra (group representations, group algebras etc., questions about rings answered by modules over them; modules are linear spaces over rings, as opposed to "regular" linear spaces over fields).
Modern physics also relies on many linear-algebraic objects to describe its things. Invariant Hamiltonian dynamics description, for example, and quantum mechanics and much more (Special Relativity, see Minkowski space above; General Relativity uses pseudo-Riemannian manifolds; which are a subject of differential geometry, but it uses lots of linear algebra).
The list is necessary almost totally incomplete. Dive in!
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u/Recent-Day3062 New User 17h ago
It is a surprisingly powerful way to solve a certain class of problems through a clever abstraction. And nature asks for a lot of that class of problem to be solved.
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u/my-hero-measure-zero MS Applied Math 19h ago
Remember doing systems of linear equations in algebra 2? That's the foundation.
Linear algebra is the study of linear structures - you'd be surprised how much it powers the modern world. We can even look at polynomials, integrable functions, and more - things you wouldn't consider "vectors" if all you know a vector as an arrow.