Linear Transformation Equality

In summary: Let w exist in R(T*) => w is a linear combination of columns of T*=> w is a linear combination of columns of T*T=> w is in R(T*T)Since w is arbitrary, this holds for all w and so R(T*) is a subset of R(T*T)Therefore, R(T*T) = R(T*)In summary, we can prove that the range of T*T is equal to the range of T* by showing that each is a subset of the other. This can be done by recognizing that the range is given by the span of the column vectors, and then considering T*u for an arbitrary vector u. By invoking the fact that the dimension of the column and row space are the same
  • #1
jnava
7
0
Hi, I am trying to prove the following equality

Range(T*T) = Range(T*)
where T is a linear transformation and * denotes the adjoint.

I know I must first show that Range(T*T) Range(T*) and vice versa.

so, Let w exist in R(T*T), then there exists a v in vector space V s.t.
T*T(v) = w.
Then I draw a blank, what's next? Or am I even starting it correctly?

Thanks
 
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  • #2
how about something like this as a strawman:

first recognise that range is given by the span of the column vectors

now consider T*u, for an arbitrary vector u. The results is a linear combination of the vectors in T*

So the columns of T*T can be thought of as linear combinations of the columns of T*

Then you probably need to invoke that the dimension of the column and row space are the same to finish
 
  • #3
I am slowly pounding through it. This is what I have:

Let w exist in R(T*T), then there exists a v in V s.t.

T*T(v) = w
=> T(v) exists in R(T*) => w exists in R(T*) => R(T*T) is a subset of R(T*)

The other way is trickier
 

What is a linear transformation equality?

A linear transformation equality is a mathematical expression that shows the relationship between two linear transformations. It is used to determine if two linear transformations are equivalent or if one can be transformed into the other by a combination of scaling, rotating, reflecting, or shearing.

How do you know if two linear transformations are equal?

Two linear transformations are equal if and only if they produce the same output for all possible inputs. This means that for every vector x in the domain, the output of the first transformation must be equal to the output of the second transformation.

What is the difference between linear transformation equality and matrix equality?

Linear transformation equality compares the outputs of two linear transformations, while matrix equality compares the elements of two matrices. Matrices can represent linear transformations, but not all matrices are equal, even if they represent the same linear transformation.

Can two linear transformations with different matrices be equal?

Yes, two linear transformations can be equal even if their matrices are different. This is because matrices can be transformed into equivalent forms by multiplying by an invertible matrix. The resulting matrices may look different, but the transformations they represent are equal.

What is the significance of linear transformation equality in science?

Linear transformation equality is an important concept in many scientific fields, particularly in physics and engineering. It allows scientists to compare different transformations and understand how they affect a system. It is also used in data analysis and computer graphics to transform and manipulate data efficiently.

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