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What is Linear Algebra?


Date: 09/06/97 at 07:26:23
From: wchoy
Subject: College level

I was wondering if you could help me with College level maths. 
I attend a University in Australia and I have absolutely no idea 
about linear algebra. To illustrate this, the lecturer has been 
talking about kernels and dimensions and from what I gather, a kernel 
is associated with corn and a dimension is on of three that we exist 
in.

ANY sort of help would be greatly appreciated!

Warren Choy


Date: 09/12/97 at 12:04:42
From: Doctor Rob
Subject: Re: College level

Linear algebra is the study of vector spaces and of linear maps 
between them.

A vector space over a field F is a set of elements V and two 
operations. 

The first operation is "plus," and it maps any pair of elements in V 
to an element of V. V with the operation "plus" is an abelian group.  

The second operation is "times," and it maps a pair consisting of an 
element of F and an element of V to an element of V.  "times" is 
associative, distributes over "plus" in V and + in F, and has the left 
identity the element 1 in F.

An example is V = the set of all n-tuples of real numbers, F = the 
field of real numbers.  "plus" is coordinate-wise addition, i.e.,
((x1,...,xn),(y1,...,yn)) |--> (x1+y1,...,xn+yn).  "times" is defined 
by scalar multiplication:  [a,(x1,...,xn)] |--> (a*x1,...,a*xn).

Every vector space has a "basis," a set of elements {v(i)} such that

  (1) they are linearly independent, i.e., if Sum x(i)*v(i) = 0, and
      each x(i) is in F, then each x(i) = 0; and

  (2) every element in V can be written in the form Sum x(i)*v(i), 
      where every x(i) is in F.

It turns out that every basis has the same size.  The number of
elements in any basis is called the dimension of the vector space.

A linear map from one vector space V to another W is a function
T: V --> W such that for any a and b in F and any x and y in V,

   T(a*x+b*y) = a*T(x)+b*T(y).

If you know T(v(i)) for any basis, you know what T does to any element
of V.

The kernel of a linear map T is the set of all v in V such that
T(v) = 0.  It turns out that this is a subset of V which is a vector
space using the same "plus" and "times" as are defined for V.  Such
things are called vector subspaces.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Linear Algebra

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