Linear Algebra's purpose in a nutshell

[torquemada]

hey i was wondering if you can define it really quick in terms of practical applications. The way calculus is so important b/c it allows us to study change and analyze different snapshots in time of change. what makes LA so epic for engineering and science - what does it enable us to do that is so important - thx

 

[lurflurf]

Linear models are very common and also very simple. Linear algebra tells us how to deal with such models. Sometimes we use linear models directly, other times we use them to approximate a more complicated model, and sometimes we use the linear model as a base and build another model on top. Calculus is mostly about ussing limits to replace a nonlinear model with a linear one, integration and differentiation are both ways to linearize a non linear function, linear algebra shows us why we wanted a linear approximation annd what to do with it once we have it.

 

[epenguin]

Polynomial algebra sounds like it would be the antithesis of linear. Instead, as a polynomial is linear in the coefficients a lot of it, e.g. discriminants and other invariants, is really linear.

 

[Metaleer]

Also, the theory of linear differential equations is firmly rooted in Linear Algebra. Remember that the general solution of an nth-order linear homogeneous equation is a linear combination of n linearly independent solutions... which is exactly how one obtains a whole vector (sub)space, adding up linear combinations of the basis elements.

The nonhomogeneous case isn't any different, as this time, we have to add a particular a solution to the general solution of the homogeneous equation. This is pretty much the same structure as that of an affine subspace (aka linear manifold),

$$S = \mathbf{p} + W,$$​

where $$W$$ is a vector subspace and $$\mathbf{p}$$ is any element of the original affine space. Here, $$\mathbf{p}$$ acts as the particular solution of the original nonhomogeneous equation, and $$W$$ is the general solution of the homogeneous equation.

Furthermore, the idea of linear mappings is heavily used in Calculus: for instance, to give a proper meaning to differentials, or as an indirect way to define a differentiable function, by requiring the existence of a certain linear mapping.

 

[FourierFaux]

Linear Algebra provides most of the underlying theory that allows us to move an image matrix across the screen via transformations.

An example:

Let's say that I have a strange shape that I've defined. (More than likely it was created with respect to an origin). You want to move the shape on the screen.


( 1 , 0 , 0 , a ) ( x ) ( x+a )
( 0 , 1 , 0 , b ) ( y ) = ( y+b )
( 0 , 0 , 1 , c ) ( z ) ( z+c )
( 0 , 0 , 0 , 1 ) ( 1 ) ( 1 )

And anytime you want to warp an image.

What if you wanted to slam a 3D image into a 2D space?
There's a linear transformation for that.

What if you wanted to rotate a 3D image?
You can do that too.

What if you're working with ephemeral data, but you prefer cartesian coordinates?
Just slam that puppy into a Jacobian matrix and crank out your x, y and z.

Want to see what the crab nebula would look like traveling at .97c?
There's a linear transformation for that too.

This is how we use Linear Algebra. This is where all of those 12 page long proofs about linear vector spaces lead.

 

[hotvette]

http://www.math.ucdavis.edu/~daddel/linear_algebra_appl/Applications/applications.html