Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

** Every vector space has a basis.

** The closed unit ball of the dual of a normed vector space over the reals has an extreme point.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

The relatively recent novel vector has facilitated a far more rapid spread than the simple expansion of habitats North through global warming.

The yellow particles leave 5 blue trails of random motion and one of them has a red velocity vector.

An array processor or vector processor has multiple parallel computing elements, with no one unit considered the " center ".

The notation ∇ × F has its origins in the similarities to the 3 dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if we take ∇ as a vector differential operator del.

* In a vector field describing the linear velocities of each part of a rotating disk, the curl has the same value at all points.

The collection of convex subsets of a vector space has the following properties:

Each sender has a different, unique vector v chosen from that set, but the construction method of the transmitted vector is identical.

In two dimensions the position vector which has magnitude ( length ) and directed at an angle above the x-axis can be expressed in Cartesian coordinates using the unit vectors and:

The rotation itself is represented by the angular velocity vector Ω, which is normal to the plane of the orbit ( using the right-hand rule ) and has magnitude given by:

which, by properties of the vector cross product, has magnitude rdθ and is in the direction tangent to the circular path.

Because u < sub > ρ </ sub > is a unit vector, its magnitude is fixed, and it can change only in direction, that is, its change du < sub > ρ </ sub > has a component only perpendicular to u < sub > ρ </ sub >.

In mathematics, any vector space, V, has a corresponding dual vector space ( or just dual space for short ) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors.

The source-free part, B, can be similarly written: one only has to replace the scalar potential Φ ( r ) by a vector potential A ( r ) and the terms −∇ Φ by +∇× A, and finally the source-density

The determinant provides important information when the matrix is that of the coefficients of a system of linear equations, or when it corresponds to a linear transformation of a vector space: in the first case the system has a unique solution if and only if the determinant is nonzero, while in the second case that same condition means that the transformation has an inverse operation.

In other words, for any N > 0, an N-dimensional complex vector has a DFT and an IDFT which are in turn N-dimensional complex vectors.

The angular momentum of a particle or rigid body in rectilinear motion ( pure translation ) is a vector with constant magnitude and direction.

A special case is the inner product of a vector with itself, which is square of its norm ( magnitude ):

It is defined as a vector whose magnitude is the electric current per cross-sectional area.

where T is the magnitude of T and u is the unit tangent vector.

In other words, the apparent velocity in the rotating frame is altered by the amount of the apparent rotation at each point, which is perpendicular to both the vector from the origin and the axis of rotation and directly proportional in magnitude to each of them.

Vector relationships for uniform circular motion ; vector Ω representing the rotation is normal to the plane of the orbit with polarity determined by the right-hand rule and magnitude dθ / dt.

where vertical bars |...| denote the vector magnitude, which in the case of r ( t ) is simply the radius R of the path.

The direction and magnitude of such distortion is expressed in terms of a Burgers vector ( b ).

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar.

If y is a point where the vector field v ( y ) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude.

As the electric field is defined in terms of force, and force is a vector, so it follows that an electric field is also a vector, having both magnitude and direction.

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase.

where the angular rotation of the frame is expressed by the vector Ω pointing in the direction of the axis of rotation, and with magnitude equal to the angular rate of rotation Ω, symbol × denotes the vector cross product, vector x < sub > B </ sub > locates the body and vector v < sub > B </ sub > is the velocity of the body according to a rotating observer ( different from the velocity seen by the inertial observer ).

The coordinates of the vector are equal to the projections of the vector ( yellow ) onto the x-component basis vector ( green )-using the dot product ( a special case of an inner product, see below ).

The coordinates of the vector are equal to the projections of the vector ( yellow ) onto the x-component basis vector ( green )-using the inner product ( see below ).

The resultant or net force on the ball found by vector addition of the normal force exerted by the road and vertical force due to gravity must equal the centripetal force dictated by the need to travel a circular path.

The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalar-valued function:

The divergence of the curl of any vector field ( in three dimensions ) is equal to zero:

In statistics, a bivariate random vector ( X, Y ) is jointly elliptically distributed if its iso-density contours — loci of equal values of the density function — are ellipses.

The Dyson series can be alternately rewritten as a sum over Feynman diagrams, where at each interaction vertex both the energy and momentum are conserved, but where the length of the energy momentum four vector is not equal to the mass.

The principle of superposition of waves states that when two or more waves are incident on the same point, the total displacement at that point is equal to the vector sum of the displacements of the individual waves.

This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket, and it is equal to twice the one defined through left-invariant vector fields.

Such a vector field serves to define a generalized ordinary differential equation on a manifold: a solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.

* Unit vector, a vector with length equal to 1

Specifically, if F is a vector space of linear functionals on X which separates points of X, then the continuous dual of X with respect to the topology σ ( X, F ) is precisely equal to F.

For every vector space there exists a basis ( if one assumes the axiom of choice ), and all bases of a vector space have equal cardinality ( see dimension theorem for vector spaces ); as a result the dimension of a vector space is uniquely defined.