Not always. Any m by n matrix is also a vector. Polynomials are vectors. As are continuous functions.
A vector is an element of a vector space over a field. These are sets which have a few operations, vector addition and scalar multiplication, and obey some well known rules, such as the existence of a zero vector (identity for vector addition), associativity and commutativity of vector addition, distributivity of scalar multiplication over vector sums, that sort of thing!
These basic properties give rise to more elaborate concepts such as linear independence, spanning sets, and the idea of a basis, though not all vector spaces have a finite basis.
Every vector is a tensor. Matrices are vectors because m by n matrices form vector spaces. Magnitude and direction have nothing to do with the definition of vectors which are just elements of vector spaces.
All vectors are tensors but not vice versa. And every page/definition of vector I've seen references magnitude and direction, even the vector space page you linked.
It looks like "vector" commonly refers to geometric vectors which is what most folks in this thread are discussing.
Would N by M vectors be imaginary, where each DOF has real and imaginary components?
Continuous functions on [0,1] are vectors. Magnitude and direction are meaningless in that vector space, usually called C[0,1]. Magnitude and direction are not fundamental properties of vectors.
n by m matrices (and the vector spaces to which they belong) are perhaps best thought of similarly to functions and function spaces. Not as geometric objects, but as linear transformations (which they are).
Your polynomial, f(x) = a + bx +cx^2 + dx^3, is an element of the vector space P3(R), the polynomial vector space of degree at most 3 over the reals. This space is isomorphic to R^4 and it has a standard basis: {1, x, x^2, x^3}. Then you can see that any such f(x) may be written as a linear combination of the basis vectors with real valued scalars.
As an exercise, you can check that P3(R) satisfies some of the properties of vector spaces yourself (existence of zero vector, associativity and commutativity of vector addition, distributivity of scalar multiplication over vector sums).
What happens to elements with powers of x above 3? Say we multiply the example vector above with itself. We would end up with a component d^2^x^6^, witch is not part of the P3R vector space, right?
Do we need a special multiplication rule to handle powers of x above 3?
I've worked with quaternions before, which has " special" multiplication rules by defining i j and k.
That's only if you're working with the perspective of it being a polynomial. When you're considering the polynomial as a vector however, that operation simply doesn't exist
Multiplication of two vectors is not an operation defined on vector spaces. If you want that, you’re looking at either a structure known as an inner product space or an algebra over a field.
Note that the usual notion of polynomial multiplication doesn’t apply to polynomial vector spaces, nor does it agree with the definition of an inner product nor the bilinear product of an algebra.
It's not a terrible name, since it's derived from the mathematical construct of vectors as n-tuples. In the case of vectors in programming, n relates to the size of the underlying array, and the tuple consists of the elements of the vector.
The only correct answer for a 101 introduction. It’s an incredible powerful intuition even in contexts where vectors are seemingly used as a list of numbers.
You can also define a vector by the equivalent “sides of the right triangle”. In 2D, the x,y coordinates. In computer science, vectors are n-tuples, so they represent a math/physics vector but in n-dimensions.
But the vector space of (all) real functions is a completely different beast from the space of computable functions on finite-precision numbers. If you restrict the equality of these functions to their extension,
defined as f = g iff forall x\in R: f(x)=g(x),
then that vector space appears to be not only finite dimensional, but in fact finite.
Otherwise you probably get a countably infinite dimensional vector space indexed by lambda terms (or whatever formalism you prefer.)
But nothing like the space which contains vectors like
It's a dynamically-sized list of objects of the same type stored contiguously in memory.
dynamically-sized: The size of it can change as needed.
list: It stores multiple things together.
object: A bit of programmer defined data.
of the same type: all the objects in the list are defined the same way
stored contigiously in memory: if you think of memory as a bookshelf then all the objects on the list would be stored right next to each other on the bookshelf rather than spread across the bookshelf.
Dynamically sized but stored contiguously makes the systems performance engineer in me weep. If the lists get big, the kernel is going to do so much churn.
matlab likes to pick the smallest available spot in memory to store a list, so for loops that increase the size of a matrix it's recommended to preallocate the space using a matrix full of zeros!
Preallocate the vector if you can guesstimate the size
Use a vector library that won't reallocate the entire vector on every single addition (like Rust, whose Vec doubles in size every time it runs out of space)
Memory is fairly cheap. Allocation time not so much.
Its the algebraic properties that are important, not all vectors are n-tuples, eg the set of polynomials of degree less than n.
You need a basis to coordinate a vector, you can work with vectors without doing that and just deal with the algebraic properties. The coordinate representation is dependent on the basis chosen and isn't fundamental to the vector. So calling them n-tuples isn't technically correct.
You can turn them into a set of coordinates if you have a basis, but the fact that you can do that is because of the algebraic properties so it's those properties which define what a vector is.
I think a better example to show how vectors don't necessarily need to be what people conceptualize as n-tuples would have been the real numbers. (Of course, these can be considered 1-tuples, but the same can be said of any arbitrary set element that is not itself a tuple with more entries.) A cooler example would have been R[x] (the ring of real-valued polynomials of a single variable) especially since an isomorphic ring using n-tuples would be a more cumbersome representation of the algebra.
No. ArrayList is thread safe and implements the connections API. Vector doesn't. Though if you're using Java, there's almost no instance where you would want to use a Vector instead of ArrayList.
Only if one thread modifies it while another one is iterating over it, if two threads try to modify the list at once there isn't any kind of synchronization and it really could break your list.
