Exploring Linear Algebra: Basics, Uses, and Applications

[The_Z_Factor]

Hi, I was wondering this, because I actually never heard of linear algebra until I came to this site (which makes me believe that linear algebra is a college level math because I'm pretty sure I haven't seen it as a course at my high school), what is linear algebra? What is it used for?

 

[The_Z_Factor]

I looked it up on wikipedia and found a lot of complicated looking equations, so if you're wondering that's why I'm asking you guys. Hoping for a simple explanation :tongue2:

 

[salman213]

If ur in Canada you will begin learning basics of linear algebra in Grade 12 (high school) according to last years curriculum, BUT i heard last year (when i was in high school) that they were combining linear algebra and calculus as vector calculus in gr 12. So if that's the case you probably won't know what it is, specifically, until uni or college. Basically there are so many braches of it that its hard to give a concrete definition from what I know (which is not much :D)

"Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations."

 

[salman213]

i forgot to add its pretty hard sometimes :)

 

[D H]

what is linear algebra? What is it used for?

The use of linear algebra is so widespread that better question is what areas of science and engineering don't use linear algebra. It is a very short list.

A few completely different examples of places where linear algebra plays an essential role:

• Engineers who wants to making a vehicle go where they want it to go, point where they want it to point use linear algebra in the vehicle's guidance, navigation, and controls.
• Industrial managers who determine what mix of manufacturing products should be produced use linear algebra to find the mix that maximizes profit.
• Scientists who have a mathematical model of some process and some experimental observations of the process use linear algebra to deriving the coefficients for the model.
• Safety analysts use linear algebra to determine what loads will make a bridge unsafe.

I just picked a few examples off the top of my head. Others can add to the list. Bottom line: Linear algebra is ubiquitous throughout science and engineering.

 

[The_Z_Factor]

Perhaps it'd be easier for me to understand if I knew how to apply math to vectors...

So, what's vector calculus? Is that basically physics..I remember (before I stopped trying to learn physics on my own due to lack of math education :tongue:), I was doing vectors and there were some problems like, "If a river moves south at 5.0 m/s, and you have a boat that you know can move 4.6 m/s in still water, what is the velocity and angle you need to get across the river?" I don't think it was said like that, I don't remember, but basically you used the pythagorean theorem to figure out "c" the hypotenuse, which was the "route" the boat was taking (in this question), and you'd come up with an answer something like, "x degrees y velocity North of West". I think.

Anyways, is it something like that?

 

[HallsofIvy]

I think it would be more correct to say that Linear Algebra is the study of Finite dimensional vector spaces (in fact, that is the title of Halmos' classic textbook).

The Z Factor, Linear Algebra is a mathematical generalization of that. It's real importance is that it incapsulates all of the properties that "linear" problems have.

 

[mathwonk]

its what all freshmen think all math is likje. i.e. if you add the inputs the outputs also add.

so if square roots were linear, then sqrt(a+b) would be sqrt(a)+srt(b), as my students wish were true.

i.e. linearity means f(a+b) = f(a)+f(b).

 

[ROLEX4life]

Perhaps it'd be easier for me to understand if I knew how to apply math to vectors...

So, what's vector calculus? Is that basically physics..I remember (before I stopped trying to learn physics on my own due to lack of math education :tongue:), I was doing vectors and there were some problems like, "If a river moves south at 5.0 m/s, and you have a boat that you know can move 4.6 m/s in still water, what is the velocity and angle you need to get across the river?" I don't think it was said like that, I don't remember, but basically you used the pythagorean theorem to figure out "c" the hypotenuse, which was the "route" the boat was taking (in this question), and you'd come up with an answer something like, "x degrees y velocity North of West". I think.

Anyways, is it something like that?

Not exactly. The problem you describe is just a basic application of vector arithmetic. Vector Calculus mainly deals with integration in a multi-dimensional Cartesian space (3 dimensions or higher). It is in a way related to Multi-Variable Calculus. In Single Variable Calculus, you integrate functions of one variable over a segment of the real number line (for example, integrate f(x) = ln(x) from x = 2 to x = 7). In Vector Calculus, you generalize that idea to accommodate all possible situations. For example, what if you are integrating a function of 2 variables z = f(x,y) over a line in the x-y plane described by y = f(x) or as a set of parametric equations x = f(t) and y = g(t) where "t" (usually interpreted as time) is the parameter? Situations like that get more complicated and require the concept of vectors. That example I just described is a "line integral" (or "path integral") and there are other such types of integrals which require vectors such as surface/flux integrals. In real life, sometimes you have to integrate over Vector fields and other such mathematical entities that are not as simple as a part of the real number line and so Vector Calculus provides us with a way to do that.

 

[rdt2]

Linear algebra is often just another name for matrix algebra.

 

[radou]

As HallyofIvy already pointed out, linear algebra is the study of finite-dimensional vector spaces. Another important fact is that mappings between such algebraic structures are often the most interesting part.

 

[HallsofIvy]

Linear algebra is often just another name for matrix algebra.

Ouch! Ouch! You are not far from right but one the most difficult things in teaching Linear Algebra is convincing the students that "Linear Algebra" and "matrix algebra" are NOT the same thing.

It is true that a vector in any finite dimensional vector space, , after you have chosen a specific basis, can be written as a member of Rⁿ, just by writing the coefficients of of the vector written as a linear combination of the basis vectors. It is also true that any linear transformation from one vector space to another, after you have chosen a specific basis for each vector space, can be written as a matrix. But it is crucially important that students understand that those a "representations" of the vectors and linear transformations, not the "things themselves"! In particular you must keep in mind the fact that all of these depend upon the choice of specific bases. Choosing different bases for the vector spaces, you get a completely different matrix representing the same linear transformations. One of the thing you should learn in Linear Algebra is how to change from one representation to another.

 

[mathwonk]

try my website, free notes on linear algebra.