Why is Linear Algebra easier to understand than Calculus?

[KingKai]

I had a linear algebra course for my 1st year civil engineering curriculum, and I passed with a 3.2 GPA but I only conceptually understood about 10% of what was taught to me.


I don't know what an eigenvalue/eigenvector is, what the hell is a subspace, nullspace, imagespace. What the hell is a linear transformation, what the hell is a determinant of an nxn matrix, what the hell is a matrix.


How the hell was I able to get a decent mark in a subject I know nothing about?

Facepalm.


I found calculus 1 (single variable) way easier to understand than this stuff.

Why is linear algebra so important?

Enlighten me.


My future math courses are

Calculus 2
Calculus 3
Differential Equations
Partial Differential Equations
Advanced Mathematics (Whatever the hell that is)

 

[Vargo]

I can't explain the whole linear algebra curriculum in a short post, but it is a fundamental part of mathematics. It sounds like you basically know nothing about the subject but you managed to pass with a decent grade. Good job?

1. Solving systems of linear equations is part of linear algebra, and this is probably the part that is used the most by the most people.

2. Finish calculus and differential equations and then revisit linear algebra. Conceptually it might make more sense then.

 

[tahayassen]

Yeah, I don't understand it either. I scored well on tests, but I didn't understand half of it. Just memorized the theorems and applied them on the test. Problem is that they went way too fast for something that we're learning for the first time.

 

[Stephen Tashi]

Yeah, I don't understand it either. I scored well on tests, but I didn't understand half of it. Just memorized the theorems and applied them on the test. Problem is that they went way too fast for something that we're learning for the first time.

If you want to understand linear algebra intuitively, you'll get better advice by asking for an explanation of one concept at a time rather than by asking for an explanation of the entire subject. When it comes to intuition, people have a wide variety of ways of looking at mathematical concepts. You can get 5 different ways of looking at one simple idea.

 

[johnqwertyful]

The key with linear algebra is mathematical maturity. You need to understand that definitions are just definitions. There's nothing deeper.

An eigenvalue, λ, and eigenvector, x of a matrix A are such that Ax=λx. That's IT. There's absolutely nothing more to it than that. That's all that it means. Why you care, how it's used is a completely different question. But that's all it is.

A matrix is just an array of numbers. That's ALL. Nothing more. That's all there is to it. Nothing deeper, nothing more. An array of numbers. Don't try to pull things out of it that simply aren't there. Yes, you can do cool things with it. Yes, you apply it in weird places. But that's ALL IT IS. An array of numbers.

The one thing I think that's taught poorly is vector spaces. Why they give you an example of an algebraic structure before you understand what an algebraic structure IS, is completely past me.

An algebraic structure is a SET with ONE OR MORE operations defined on it. In a VECTOR SPACE, the set is the set of vectors. The operations are scalar multiplication and vector addition.

An algebraic structure IS math. It's such a confusing, deep subject if you don't really understand what's going on. But when you get it, it's pretty cool. Anything you do in math is in an algebraic structure (most the time, you're dealing with Euclidean space. Euclidean space is the "normal" space with "normal" rules).

A much better example of a structure is what's called a FIELD. (NOT a vector field, when you get to multivariate calculus. This is extremely important) A FIELD is a structure with elements that has two operations, + and * defined over it. It has a list of axioms; closure, 4 additive ones, 4 multiplicative ones, one associative one. An axiom is a DEFINITION.

See, in the real world we don't have wild 2s running around. "2" does NOT exist in nature. You always have 2 something. 2 rocks, 2 buildings, 2 blades of grass, 2 whatever. But "2" does NOT exist. We CREATE "2" to describe the real world. To describe the world, we create these ALGEBRAIC STRUCTURES.

A field IS numbers. When you ask your friend what 2+2 equals, you're working in a FIELD, namely R (the real numbers). Make sense? An algebraic structure IS math. Whatever you do in math is a structure. A vector space is ANOTHER example of a structure. Just one that's studied extensively in linear algebra. Anything you want to know about the operations (EXCEPT WHAT THE OPERATIONS ARE ACTUALLY DOING!), you can derive from the axioms. In a vector space, you can derive ALL you want to know about scalar multiplication or vector addition from the axioms. BUT the one thing you CAN'T derive is WHAT YOU'RE ACTUALLY DOING when you add vectors. YOU must define that.

So long story short, you probably DO understand it. You're just looking for something that's not there. Definitions are just definitions.

 

[Vargo]

I would suggest that the above poster's characterization of linear algebra as a meaningless system of definitions and rules is precisely why the OP was not able to understand any of the concepts. Without any context, it can be hard for the brain to hold on to so much empty structure.

 

[johnqwertyful]

I would suggest that the above poster's characterization of linear algebra as a meaningless system of definitions and rules is precisely why the OP was not able to understand any of the concepts. Without any context, it can be hard for the brain to hold on to so much empty structure.

Linear algebra isn't meaningless at all, when did I ever say anything like that? You're just learning things rigorously, without much if any physical intuition.

Like with eigenvalues/eigenvectors. There really isn't a physical intuition behind it (maybe there is? I just never heard of any). It just is. That doesn't mean that it's "meaningless". It's used to solve differential equations later, which renders them super useful.

Not everything in math has some physical significance.

 

[Vargo]

I apologize for misinterpreting your comment.

I amend my comment to say that without context, an abstract system of rules and definitions such as linear algebra can be hard to hold onto.

Not everything in math has some physical significance.

I know at least one other poster here who would agree with this. I tend to disagree though.

In the case of an eigenvector, its physical significance is that it represents a subspace that is invariant under a linear transformation. The eigenvalue is the scaling factor applied to that invariant subspace. (geometry is physical enough for me )

 

[johnqwertyful]

I amend my comment to say that without context, an abstract system of rules and definitions such as linear algebra can be hard to hold onto.

I'll agree with that for sure. It takes awhile to get used to definitions. Trying to find context is its own task. Applied linear algebra is a huge (fascinating) field.

 

[Studiot]

I'm sorry to to a civil engineer with such an attitude.

Linear alegbra is vitally important to some aspects of civil engineering such as the ability to solve (very) large sets of simultaneous equations for structural or hydraulic purposes.

Knowledge of eigenvectors help prevent such disasters as Tacoma Narrows.

If concrete is important to you then so is the knowledge of the Bogue equations, which form a stochiometric linear algebra.

Perhaps soil mechanics is your bag - well linear analysis predominates (effect A is propotional to property B)

I can honestly say that the only major non linear analysis I was involved in, during my time in engineering maths in civils, concerned pressure fluctuations in major gas pipelines.

 

[chiro]

It just extends the idea of linearity to multi-dimensional objects.

It's not intuitive because you represent an object that is multi-dimensional but is treated like a single object instead of a collection of other objects.

To see this, you should look at the multi-variable and manifold calculus and look at the analogues between the one-dimensional and the multi-dimensional forms.

Linearity is the simplest kind of object and the algebra also provides techniques of decomposition and re-composition of general vectors, matrices, and even tensors.

 

[symbolipoint]

Yeah, I don't understand it either. I scored well on tests, but I didn't understand half of it. Just memorized the theorems and applied them on the test. Problem is that they went way too fast for something that we're learning for the first time.

Since you, according to one of your recent posts, are currently studying Algebra 1, you are not yet ready to understand much of Linear Algebra. Give yourself about 2 more years.

 

[KingKai]

I'm sorry to to a civil engineer with such an attitude.

I'm not a civil engineer yet, I'm an undergrad student.

Attitude? I was simply expressing my confusion over this newly (I can't stress the word newly enough) learned subject. It is my sole intention to strengthen my intuition with the subject in the same way I am intuitive with calculus and geometry.

I don't hate linear algebra, it not as though I want to attack it with a light saber, I just find it more abstract than any other branch of math I have been exposed to.

Perhaps soil mechanics is your bag

As I am a 1st year undergrad student with no exposure whatsoever to the specialties of civil engineering, drawing such conclusions based on the limited info and limited time of exposure I have had with linear algebra (3 months) is a little too extreme.

You just fueled me only to pursue linear algebra further.

Merry Christmas :)

 

[BraneChild]

Funny,
I thought Linear Algebra was easier to grasp than Calculus. I guess it's because it's hard for me to visualize a mathematical concept (it took me a while to understand what a derivative is from a geometric point of view). With linear algebra you just take a system of linear equations, strip the constants and coefficients from it and viola, you have a matrice! And from there you can apply elementary row operations on it to get a solution, find it's inverse, it's determinant, etc...

To be fair though I learned Linear Algebra independently (which probably made it easier), and I've only gotten the basics (I haven't learned about eigenvalues or linear transformations yet).