Properties
| 2↕ | = | ‹0 | |||
| ↔ 1 |
^ 0 |
↔ 1 |
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| Element of the zero space, written as empty column vector (rightmost one), is multiplied by 2×0 empty matrix to obtain 2-dimensional zero vector (leftmost). Rules of matrix multiplication are respected. |
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The trivial ring, zero module and zero vector space are zero objects of the corresponding categories, namely Rng, R-Mod and VectR.
The zero object, by definition, must be a terminal object, which means that a morphism A → {0} must exist and be unique for an arbitrary object A. This morphism maps any element of A to 0.
The zero object, also by definition, must be an initial object, which means that a morphism {0} → A must exist and be unique for an arbitrary object A. This morphism maps 0, the only element of {0}, to the zero element 0 ∈ A, called the zero vector in vector spaces. This map is a monomorphism, and hence its image is isomorphic to {0}. For modules and vector spaces, this subset {0} ⊂ A is the only empty-generated submodule (or 0-dimensional linear subspace) in each module (or vector space) A.
Read more about this topic: Zero Object (algebra)
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)