Tensors are generalizations of scalars (that have no indices), vectors (that have exactly one index), and matrices (that have exactly two indices) to an arbitrary number of indices.
But a grid can just be a number with a list of numbers. A tensor is just two numbers with a list of numbers. A n-tensor is just two lists of numbers. Two lists can be combined with a number to indicate when they split. If we put that number at the start of the list, then we just have a list.
All the geometric definitions of tensors I have met always assumed a base, such that a change of coordinate or of parametrization would change the values of the tensor. Unless you define the tensor by its action instead of its values?
Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.
That's exactly correct. It's similar to how a vector in R^2 is just an arrow with a magnitude and a direction. When you represent that arrow in different bases, the arrow itself isn't changing, just the list of numbers you use to represent them. Likewise, tensors do not change when you change bases, but their representations as n dimensional grids of numbers do change.
Which is really a roundabout way of saying a tensor is a multilinear relationship between arbitrary products of vectors and covectors. They're inherently geometric objects that don't depend on a choice of coordinate system. The box of numbers is just one way of looking at a tensor, like a matrix is to a linear transformation on a vector space
You lost me at vectors not having to be linear. You can apply nonlinear functions or operations to vectors, but doing so transforms them into a different, non-linear context, afaik. We might be using different definitions of some of these terms, especially "linear".
You just used "tensor" to define "tensor," but also any list of number formulated as an n-dimensional matrix will satisfy this criterion. A tensor is both a linear transformation and an n-dimensional box-shaped list of numbers, but there's nothing such as a linear list of numbers.
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To approach from another angle...
https://mathworld.wolfram.com/Tensor.html
That makes a lot of sense
Avengers Endgame is just a stream of numbers
All of creation is just a stream of numbers
And we call that physics
And that is beautiful
Your mom is an endless stream of numbers
420.69696969....
There is no spoon.
Excuse me, but a tensor is actually a blob of numbers which extends the concept of a matrix to a generic sequence and stride data structure.
Caught one!
A tensor is one that transforms like a tensor 🤯
You really did a number on this one
He's doing numbers!
Can you all keep it down! You’re mathing too loudly for this time of night. Some of us have to get up early(ish).
You're quite a number
But a grid can just be a number with a list of numbers. A tensor is just two numbers with a list of numbers. A n-tensor is just two lists of numbers. Two lists can be combined with a number to indicate when they split. If we put that number at the start of the list, then we just have a list.
Everything is just a list of numbers.
I am THE NUMBER
A scalar is just a 0th order tensor
A tensor is a special box of numbers that doesn't change under coordinate transformation.
Isn't a tensor the generalization of scalar, vector matrix and so on? (PLUS the invariance under coordinate transforms?)
A box would be 3-dimensional indicating that tensors have 3 indices when in reality they have n-indices. Ir am i reading it wrong?
It's an n-dimensional box
Where does this definition come from?
All the geometric definitions of tensors I have met always assumed a base, such that a change of coordinate or of parametrization would change the values of the tensor. Unless you define the tensor by its action instead of its values?
That's exactly correct. It's similar to how a vector in R^2 is just an arrow with a magnitude and a direction. When you represent that arrow in different bases, the arrow itself isn't changing, just the list of numbers you use to represent them. Likewise, tensors do not change when you change bases, but their representations as n dimensional grids of numbers do change.
Which is really a roundabout way of saying a tensor is a multilinear relationship between arbitrary products of vectors and covectors. They're inherently geometric objects that don't depend on a choice of coordinate system. The box of numbers is just one way of looking at a tensor, like a matrix is to a linear transformation on a vector space
a tensor is just an element of a tensor product. and a tensor product is just a way to multiply algebraic structures
A Pi is just an endless string of numbers.
Hnghhhhh I can't agree. Its decimal representation is an endless string of digits, yes. However, in base π, π is represented by only one digit.
3.14% of sailors are pirates.
This joke makes me sqrt(-1)%
A number is just a type of vector
Tensorflow is just Numberwang
*a subtype of vector
vector is a type not a kind :P
And this is the reason why no one knows what a tensor is. (This also completely blows up in your face as soon as you have infinite dimensions)
I mean... it also has to be linear too, but sure okay ☺️
What has to be linear? Vector?matrix? Tensor? Neither makes sense
You lost me at vectors not having to be linear. You can apply nonlinear functions or operations to vectors, but doing so transforms them into a different, non-linear context, afaik. We might be using different definitions of some of these terms, especially "linear".
"Linear" describes transformations, not numbers.
Isn't a tensor a multilinear map taking as input a tensor and outputting another tensor?
Also, but that is because math likes to reuse names like Donald Duck reuses his jacket…
You just used "tensor" to define "tensor," but also any list of number formulated as an n-dimensional matrix will satisfy this criterion. A tensor is both a linear transformation and an n-dimensional box-shaped list of numbers, but there's nothing such as a linear list of numbers.
These collections also happen to follow a few specific geometric rules.
Math is just syntax.
Overly simplified math meme is bad. Take it down