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MATH
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Definitions from Matrix representations

All definitions from Matrix representations.

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Injective map

A map T:V→WT:V\to W is injective, or one-to-one, when

T(v1)=T(v2)⟹v1=v2.T(v_1)=T(v_2)\quad\Longrightarrow\quad v_1=v_2.
  1. [1]:Linear map
  2. [2]:Injectivity and the kernel
  3. [3]:Bijective map
  4. [4]:Integration gives an injective map
  5. [5]:Bijectivity in equal dimensions

Surjective map

A map T:V→WT:V\to W is surjective, or onto, when every w∈Ww\in W equals T(v)T(v) for some v∈Vv\in V. Equivalently, im⁡(T)=W\operatorname{im}(T)=W.

  1. [1]:Linear map
  2. [2]:Bijective map
  3. [3]:Bijectivity in equal dimensions
  4. [4]:Possible nullities of a linear map
  5. [5]:Seen Questions

Bijective map

A map is bijective when it is both injective and surjective.

  1. [1]:Injective map
  2. [2]:Surjective map
  3. [3]:Inverse map
  4. [4]:Equivalent invertibility conditions

Identity map

The identity map on VV is

id⁡V:V→V,id⁡V(v)=v.\operatorname{id}_V:V\to V,\qquad \operatorname{id}_V(v)=v.
  1. [1]:Linear map

Inverse map

For a map T:V→WT:V\to W, a map S:W→VS:W\to V is an inverse of TT when

S∘T=id⁡V,T∘S=id⁡W.S\circ T=\operatorname{id}_V, \qquad T\circ S=\operatorname{id}_W.

The first composition is the left-inverse condition and the second is the right-inverse condition. An inverse satisfies both.

  1. [1]:Bijective map
  2. [2]:Equivalent invertibility conditions
  3. [3]:Inverse maps in coordinates

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Injective mapSurjective mapBijective mapIdentity mapInverse map