Linear algebra(linear transformation)

[chuy52506]

Homework Statement

Let T be a Linear Transformation from U to V with dim(V)=m where m<infinity.

Homework Equations

Show that rank(T) less than or equal to m.

The Attempt at a Solution

Im really not that familiar with what rank or m is.

 

[dextercioby]

Well, familiarity depends on your level of involvement. So look for the definition of <rank> in your notes/textbook. You can't find it (though i doubt it), search the internet: planetmath.org, wikipedia.org, wolfram sites offer a wealth of information on linear algebra.

m is the dimension of the vector space V. It's maximum number of linear independent vectors in V.

 

[chuy52506]

ok so rank is the max. number of linearly independent rows or columns.
I was trying to manipulate rank(T)=dim(range(T)) I am not sure if this is the correct starting point?

 

[dextercioby]

It is. What's range(T) equal to ?

 

[chuy52506]

No nevermind that belonged to another problem. Ok so i just have rank(T)+nullity(T)=m

 

[dextercioby]

Not really. The rank-nullity theorem http://en.wikipedia.org/wiki/Rank-nullity_theorem#cite_note-0 is different.

So once more: what's range (T) equal to ?

 

[chuy52506]

It does not say what the range(T) is

 

[dextercioby]

Of course not, you have to know that from the beginning of the class on linear algebra. The range is a subset of V obtained by taking all possible images through T of all the elements of U (assumed to be equal to the domain of T). The range is also known as codomain of a linear map/operator.

If U is a linear space and T is a linear operator, one can easily show that range(T) is also a linear space, namely a vector subspace of V.

 

[eabod]

I think the rank-nullity theorem is exactly what you want to use.