User:IssaRice/Linear algebra/Equivalent statements for injectivity and surjectivity - Revision history

IssaRice at 19:42, 30 August 2023

2023-08-30T19:42:11Z

← Older revision		Revision as of 19:42, 30 August 2023
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	Let <math>A</math> be an <math>m \times n</math> matrix.		Let <math>A</math> be an <math>m \times n</math> matrix. That's a matrix with <math>m</math> rows and <math>n</math> columns, which you can also think of as a map <math>\mathbf R^n \to \mathbf R^m</math>.

	{\| class="wikitable"		{\| class="wikitable"

IssaRice: /* External links */

2021-08-14T07:07:47Z

External links

← Older revision		Revision as of 07:07, 14 August 2021
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	* http://davidjekel.com/wp-content/uploads/2019/07/Linear_Algebra_Equivalences.pdf		* http://davidjekel.com/wp-content/uploads/2019/07/Linear_Algebra_Equivalences.pdf
			* Hubbard and Hubbard's ''Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach'' also has a similar table in section 2.5 (kernels, images, and the dimension formula).

IssaRice at 06:45, 10 January 2021

2021-01-10T06:45:04Z

← Older revision		Revision as of 06:45, 10 January 2021
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	===dimension of range = n===		===dimension of range = n===

			==External links==

			* http://davidjekel.com/wp-content/uploads/2019/07/Linear_Algebra_Equivalences.pdf

IssaRice at 05:28, 9 January 2020

2020-01-09T05:28:00Z

← Older revision		Revision as of 05:28, 9 January 2020
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	\| <math>\dim \operatorname{range} A = n</math> \|\| <math>\dim \operatorname{range} A = m</math> \|\|		\| <math>\dim \operatorname{range} A = n</math> \|\| <math>\dim \operatorname{range} A = m</math> \|\|
	\|}		\|}

			==Characterizations of injectivity==

			===left inverse===

			===Ax=b has at most one solution===

			===linearly independent columns===

			===spanning rows===

			===rank n===

			===pivot in every column===

			===null space = {0}===

			===zero-dimensional null space===

			===dimension of range = n===

IssaRice at 01:15, 14 August 2019

2019-08-14T01:15:27Z

← Older revision		Revision as of 01:15, 14 August 2019
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	\|-		\|-
	\| <math>\operatorname{null} A = \{0\}</math> \|\| <math>\operatorname{range} A = \mathbf R^m</math> \|\|		\| <math>\operatorname{null} A = \{0\}</math> \|\| <math>\operatorname{range} A = \mathbf R^m</math> \|\|
			\|-
			\| <math>\dim \operatorname{null} A = 0</math> \|\| <math>\dim \operatorname{null} A = n-m</math> \|\|
			\|-
			\| <math>\dim \operatorname{range} A = n</math> \|\| <math>\dim \operatorname{range} A = m</math> \|\|
	\|}		\|}

IssaRice at 01:12, 14 August 2019

2019-08-14T01:12:00Z

← Older revision		Revision as of 01:12, 14 August 2019
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	\| the columns of <math>A</math> are linearly independent \|\| the columns of <math>A</math> span <math>\mathbf R^m</math> \|\| the columns of <math>A</math> are a basis of <math>\mathbf R^m</math>		\| the columns of <math>A</math> are linearly independent \|\| the columns of <math>A</math> span <math>\mathbf R^m</math> \|\| the columns of <math>A</math> are a basis of <math>\mathbf R^m</math>
	\|-		\|-
	\| the rows of <math>A</math> span <math>\mathbf R^m</math> \|\| the rows of <math>A</math> are linearly independent \|\| the rows of <math>A</math> are a basis of <math>\mathbf R^m</math>		\| the rows of <math>A</math> span <math>\mathbf R^n</math> \|\| the rows of <math>A</math> are linearly independent \|\| the rows of <math>A</math> are a basis of <math>\mathbf R^n</math>
	\|-		\|-
	\| \|\| \|\| <math>\det(A) \ne 0</math>		\| \|\| \|\| <math>\det(A) \ne 0</math>

IssaRice: Created page with "Let $A$ be an $m \times n$ matrix. {| class="wikitable" ! Injective !! Surjective !! Bijective |- | $A$ is injective || $A$ is sur..."

2019-08-14T01:07:53Z

Created page with "Let <math>A</math> be an <math>m \times n</math> matrix. {| class="wikitable" ! Injective !! Surjective !! Bijective |- | <math>A</math> is injective || <math>A</math> is sur..."

New page

Let <math>A</math> be an <math>m \times n</math> matrix.

{| class="wikitable"
! Injective !! Surjective !! Bijective
|-
| <math>A</math> is injective || <math>A</math> is surjective || <math>A</math> is bijective
|-
| <math>A</math> has a left inverse || <math>A</math> has a right inverse || <math>A</math> has both a left and right inverse (which turn out to be the same)
|-
| for each <math>b</math>, the equation <math>Ax = b</math> has at most one solution (in other words, a solution may not exist, but if it does, it is unique) || for each <math>b</math>, the equation <math>Ax = b</math> has at least one solution (in other words, a solution always exists, but it may not be unique) || for each <math>b</math>, the equation <math>Ax = b</math> has exactly one (in other words, a solution always exists, and it is unique)
|-
| the columns of <math>A</math> are linearly independent || the columns of <math>A</math> span <math>\mathbf R^m</math> || the columns of <math>A</math> are a basis of <math>\mathbf R^m</math>
|-
| the rows of <math>A</math> span <math>\mathbf R^m</math> || the rows of <math>A</math> are linearly independent || the rows of <math>A</math> are a basis of <math>\mathbf R^m</math>
|-
| || || <math>\det(A) \ne 0</math>
|-
| <math>A</math> has rank <math>n</math> || <math>A</math> has rank <math>m</math> || <math>A</math> has rank <math>n=m</math>
|-
| in the row echelon form of <math>A</math>, there is a pivot in every column || in the row echelon form of <math>A</math>, there is a pivot in every row || in the row echelon form of <math>A</math>, there is a pivot in every column and every row
|-
| <math>\operatorname{null} A = \{0\}</math> || <math>\operatorname{range} A = \mathbf R^m</math> ||
|}