The term is not very clear, as it could either mean:
A common case is , and .
One thing that makes such functions particularly simple is that they can be fully specified by specifyin how they act on all possible combinations of input basis vectors: they are therefore specified by only a finite number of elements of .
As such, when we say "linear map", we can think of a generalization of matrix multiplication that makes sense in infinite dimensional spaces like Hilbert spaces, since calling such infinite dimensional maps "matrices" is stretching it a bit, since we would need to specify infinitely many rows and columns.
For the typical case of a linear form over , the form can be seen just as a row vector with n elements, the full form being specified by the value of each of the basis vectors.
Since a linear form is completely determined by how it acts on a bases, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the , and so they are isomorphic because all vector spaces of the same dimension on a given field are isomorphic, and so the dual is quite a boring concept in the context of finite dimension.
Infinite dimension seems more interesting however, see: https://en.wikipedia.org/w/index.php?title=Dual_space&oldid=1046421278#Infinite-dimensional_case
One place where duals are different from the non-duals however is when dealing with tensors, because they transform differently than vectors from the base space .
Dual vectors are the members of a dual space.
In the context of tensors , we use raised indices to refer to members of the dual basis vs the underlying basis: are defined to "pick the corresponding coordinate" out of elements of V. E.g.: Kronecker delta as:
Linear map of two variables.
More formally, given 3 vector spaces X, Y, Z over a single field, a bilinear map is a function from:
Some definitions require both of the input spaces to be the same, e.g. , but it doesn't make much different in general.
The most important example of a bilinear form is the dot product. It is only defined if both the input spaces are the same.
Proof: the value of a given bilinear form cannot change due to a change of bases, since the bilinear form is just a function, and does not depend on the choice of basis. The only thing that change is the matrix representation of the form. Therefore, we must have:
Requires the two inputs and to be in the same vector space of course.
The prototypical example of it is the complex dot product.
Same value if you swap any input arguments.
Change sign if you swap two input values.
Implies antisymmetric multilinear map.
The definition of the "dot product" of a general space varies quite a lot with different contexts.
Most definitions tend to be bilinear forms.
We use the unqualified generally refers to the dot product of Real coordinate spaces, which is a positive definite symmetric bilinear form. Other important examples include:
The rest of this section is about the case.
The default Euclidean space definition, we use the matrix representation of a symmetric bilinear form as the identity matrix, e.g. in :
An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) shows that this is a tensor that represents the volume of a parallelepiped.
It takes as input three vectors, and outputs one real number, the volume. And it is linear on each vector. This perfectly satisfied the definition of a tensor of order (3,0).
Given a basis and a function that return the volume of a parallelepiped given by three vectors , .
Name origin: likely because it "determines" if a matrix is invertible or not, as a matrix is invertible iff determinant is not zero.
Since a matrix can be seen as a linear map , the product of two matrices can be seen as the composition of two linear maps:
Every invertible matrix can be written as: real number.
Intuitively, Note that this is just the change of bases formula, and so:
- changes basis to align to the eigenvectors
- multiplies eigenvectors simply by eigenvalues
- changes back to the original basis
The main interest of this theorem is in classifying the indefinite orthogonal groups, which in turn is fundamental because the Lorentz group is an indefinite orthogonal groups, see: all indefinite orthogonal groups of matrices of equal metric signature are isomorphic.
It also tells us that a change of bases does not the alter the metric signature of a bilinear form, see matrix congruence can be seen as the change of basis of a bilinear form.
For example, consider:
Now, instead of , we could use , where is an arbitrary diagonal matrix of type: : by any positive number and . Since we are multiplying by two arbitrary positive numbers, we cannot change the signs of the original eigenvalues, and so the metric signature is maintained, but respecting that any value can be reached.
From effect of a change of basis on the matrix of a bilinear form, remember that a change of basis modifies the matrix representation of a bilinear form as:
So, by taking , we understand that two matrices being congruent means that they can both correspond to the same bilinear form in different bases.
There's a catch though: now we don't have explicit matrix indices here however in general, the generalized definition is shown at: https://en.wikipedia.org/w/index.php?title=Hermitian_adjoint&oldid=1032475701#Definition_for_bounded_operators_between_Hilbert_spaces
This set forms a ring.
Members of the orthogonal group.
Can represent a symmetric bilinear form as shown at matrix representation of a symmetric bilinear form, or a quadratic form.
The definition implies that this is also a symmetric matrix.
WTF is a skew? "Antisymmetric" is just such a better name! And it also appears in other definitions such as antisymmetric multilinear map.
- : matrix in the old basis
- : matrix in the new basis
- : change of basis matrix
The change of basis matrix is the matrix that allows us to express the new basis in an old basis:
The usual question then is: given a vector in the new basis, how do we represent it in the old basis?
That is the matrix inverse.
Every vector space is defined over a field.
Any field can be used, including finite field. But the underlying thing has to be a field, because the definitions of a vector need all field properties to hold to make sense.
Elements of the underlying field of a vector space are known as scalar.
The Wikipedia page of this article is basically a masterclass why Wikipedia is useless for learning technical subjects. They are not even able to teach such a simple subject properly there!
- https://www.maths.cam.ac.uk/postgrad/part-iii/files/misc/index-notation.pdf gives a definition that does not consider upper and lower indexes, it only counts how many times the indices appearTheir definition of the Laplacian is a bit wrong as only one appears in it, they likely meant to have written instead of , related:
TODO what is the point of them? Why not just sum over every index that appears twice, regardless of where it is, as mentioned at: https://www.maths.cam.ac.uk/postgrad/part-iii/files/misc/index-notation.pdf.
Vectors with the index on top such as are the "regular vectors", they are called covariant vectors.
Those in indices on bottom are called contravariant vectors.
It is possible to change between them by Raising and lowering indices.
Then a specific metric is involved, sometimes we want to automatically add it to products.