Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [vector-spaces]

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars

This tag is for questions about vector spaces and their properties, as well mappings between vector spaces. More general questions about linear algebra belong under the tag.

A vector space consists of a set of elements called "vectors" and is associated with a field (a set with well-behaved notions of addition, multiplication, subtraction and division) called the "field of scalars". An individual vectors can be multiplied by elements of the field of scalars to produce a new vector in the vector space, and pairs of vectors can be added or subtracted to produce a new vector as well. A full introduction can be found on Wikipedia.

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Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to $R^n$?

Timothy Gowers asks Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to $R^n$? and lists some reasons. The most powerful of these is probably There are many important examples throughout mathematics of…
TripleA
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Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism?

Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique functional in $(V \otimes W)^*$ sending $v \otimes w…
Mike F
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Proof of equivalence of algebraic and geometric dot product?

Geometrically the dot product of two vectors gives the angle between them (or the cosine of the angle to be precise). Algebraically, the dot product is a sum of products of the vector components between the two vectors. However, both formulae look…
PhD
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Does zero curl imply a conservative field?

A field that is conservative must have a curl of zero everywhere. However, I was wondering whether the opposite holds for functions continuous everywhere: if the curl is zero, is the field conservative? Can someone please give me an intuitive…
John A.
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Transpose of a linear mapping

There seems to be two kinds of transposes of a linear mapping: If $f: V→W$ is a linear map between vector spaces $V$ and $W$ with nondegenerate bilinear forms, we define the transpose of $f$ to be the linear map $^tf : W→V$, determined by …
Tim
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How can a subspace have a lower dimension than its parent space?

If $V$ is a vector subspace of $W$, then $$\dim(V) \le \dim(W)$$ Why? Does that mean that for $$W = \mathbb{R}^3\\ V = \{(0,0)\}$$ $V$ is a valid subspace of $W$? But $V$ only has two coordinates, and $W$ has $3$... I've always been under the…
Saturn
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how can a set of functions form a vector space?

I am reading a book on partial differential equations. One of the exercise question in the book is: Show that the functions $(c_1 + c_2 sin^2x + c_3 cos^2x)$ form a vector space. Find a basis of it. What is its dimension? I don't know how…
user56834
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Is there a geometric meaning to the outer product of two vectors?

Define two vectors v and u in $\mathbb{R}^3$. I know the geometric meaning of the inner and cross product. Is there a meaning to the matrix resulting from $\textbf{uv}^T$?
miha priimek
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Determining a perpendicular vector to two given vectors.

I am given two vectors: u = [0, 2, 1] v = [1, -1, 3] I need to find a vector that is perpendicular to both vectors u and v. So far, this is what I have: let n = [x, y, z] eqn 1: 0x + 2y + z = 0 eqn 2: x - y + 3z = 0 I have to use matrices to find…
Ol' Reliable
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Linear Algebra, Vector Space: how to find intersection of two subspaces?

$${ W = Sp\{{(1,3,4),(2,5,1)\}}\\ U = Sp\{{(1,1,2),(2,2,1)}} \}$$ Find a span $$U \cap W$$ First time using Math latex, pretty hard.
Ilan Aizelman WS
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Subspace generated by permutations of a vector in a vector space

Let $K$ be a field. Consider the vector space $K^n$ over the field $K$. Suppose $(a_1,a_2, ... ,a_n) \in K^n$. What is the dimension of the subspace generated by all the permutations of $(a_1,a_2,...,a_n)$? I think there are 4 different cases…
Mohan
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Examples of 'almost' vector spaces where unitary law fails

I was looking at the definition on wikipedia of a vector space (similar/equivalent definitions are everywhere, but I thought I'd list it here for completion): A vector space over a field $F$ is a set $V$ together with two binary operations that…
Abe Schulte
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A question on vector space over an infinite field

Can a vector space over an infinite field be a finite union of proper subspaces ?
user123733
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Is the empty set a vector space?

I think the empty set satisfies all of the axioms of a vector space except the one about the existence of an additive identity. Is this right?
user209799
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Color space as a vector space

I am not sure that this is the best place for this topic, so I apologize in advance. I have two questions. I think that color space with say additive colors (red, green, blue) forms a vector space. The colors add and there are inverses. First: One…
user332420
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