aka linear transformations or vector space homomorphisms
BOLD IT ALL OUT!
Used in various machine learning applications, and also in 2D and 3D animation, where the object size and shape needs to be transformed from one viewing angle to the other. An object can also be rotated and scaled within a space using geometric transformations. They are also used to project data from one space to another.
Let and be vector spaces over a field of dimensions and respectively. A mapping or transformation, , or more simply, is called a linear mapping or transformation if:
in a more succinct fashion, .
Another quick way to check a linear transformation is to see if the zero vector of maps to the zero vector of .
Examples:
- such that is a linear transformation.
- Proof: Let’s take two vectors from the vector space , and . Let’s see if this satisfies the first condition: Now
- Hence both were same. Now let’s check the second condition:
- Hence this too holds. Thus, it’s a linear mapping or transformation. You can try to do this in the “succinct form” as well and you will get the same result.
- such that is also a linear transformation.
geometrically, a linear transformation scales the vector or rotates it. Sometimes it also changes its shape.
- a linear map from T to itself is called a linear operator
- a linear map from a vector space to an underlying field is called a linear function
Matrix of a linear map
Every linear transformation has a matrix, and every matrix has a corresponding linear transformation.
Coordinates
In a vector space and an ordered basis of ; for any vector we obtain a unique representation:
of wrt . Then are the coordinates of with respect to and the vector is the vector/coordinate representation of with respect to .
Transformation matrix
Suppose and are finite dimensional vector spaces with bases and respectively. Let be a linear map. Then the matrix of with entries is defined by:
for
Hence, the matrix of (not to be confused with the transpose, no role here):
It shows that the column of the matrix of linear map consists of the coordinates in the chosen basis for .
Conversely, every matrix induced a linear map given by
example:
- Consider a linear map defined by . Find the matrix representation of relative to the basis .
- hence
- Thus,
- Consider the matrix . Find its linear map.
Kernel and range of a linear map
Let be a linear map. Then the kernel or the null space of is defined as
The range of is defined as
In plain English: the range is a collection of all vectors in the destination space such that there is at least one vector in the starting space such that the transformation turns into . In simpler terms, the range is the collection of every possible output that can produce. (Why? Think about which vectors are allowed here: If I have a random vector in the destination space and if there is no (input) such that , it never makes it into the collection. Thus the only vectors which are in the collection are where turns into that particular vector, or the outputs of .)
- Null space of and kernel of mean the same thing
- Range and image are the same thing.
- is a subspace of V.
- is a subspace of W.
- The dimension of or is called the nullity of T.
- The dimension of is called the rank of T.
Rank Nullity Theorem
For a linear map we have:
Example: determine the range and null space of the linear map defined by
Here: .
A. Determining the range.
First of all, note that the range is the set of all possible outputs of a transformation. Now it turns out that the starting space here is . To write a general vector we use the standard bases to write it. Now again it is such that the standard bases are . These “span” in the sense that we can write any vector using these. Thus we can write that . Now recall that for a linear transformation two properties are required,
Recall also that here are scalars as the general vector is but a linear combination of those basis vectors. Because we know that it’s a transformation, we can use the aforementioned properties to write it in a better form . Now all you have to do is calculate the values given the problem’s constraints! Because we have expressed every standard transformation ‘s we know that the span of vectors ‘s would form the range of this mapping.
So first we will take the standard basis in which is and see what it evaluates to. . Now take another standard basis in , and see what that evaluates to. . Now take another standard basis, and thus . We can see that the last two are just times each other. As the basis contains only linearly independent vectors, we can say that here, (note that we could’ve picked the other vector too as it’s only -1 times this vector; because it was not “unique” in the sense that it was a linear combination of some other vector, so they dropped it to avoid redundancy.)
.
B. Determining the null space or the kernel
We know that the kernel is just vectors in V such that the transformation of that vector gives a zero vector. Thus we will take the general vector in and equate it to the zero vector in (aka ).
Here, it’s . Put and thus from the first component and from second component. Also, from the third component and from the fourth component. (Note that the zero vector here is of the form .) Thus the null space would be . Here .
Note that the rank nullity theorem is verified as the dimensions of are also 3.
Basis change
Suppose you draw a dot on a wall, that dot represents a vector. Now you need something to describe the vector. You choose two basis vectors, say with basis vectors . You decide that you can describe the dot (vector) sufficiently well using these two basis vectors. But then your friend comes along and chooses different basis vectors, ex. (note that for a 2D space there can only be two basis vectors.) - now what do you do? You need something of a change of basis matrix that “translates” your standard basis to his and vice versa.
Consider for a linear mapping we have the ordered bases and for and the ordered bases and for . We use tilde here to differentiate the two bases as they are in different coordinate systems.
Consider also a transformation matrix of with respect to , and the corresponding transformation matrix with respect to . Then is given as:
where is the transformation matrix of , that maps coordinates with respect to onto coordinates with respect to , and is the transformation matrix of that maps coordinates with respect to onto coordinates with respect to .
Note that the identity mapping is defined as ; you can think of it as . See MML sec 2.7.2 for the proof.
This relation tells us that with a basis change in (when is replaced by ) and W (when is replaced by ) the transformation matrix of the given linear mapping is replaced with an equivalent matrix which satisfies the aforementioned relation.