Linear dependence: Difference between revisions

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==Rank-deficiency and full-rank==

[[File:Full-rank - updated.png|thumb|600x600px|Rank of a matrix can be revealed by reducing a matrix such as with Hermite normal form (row or column style). The shapes of the reduced matrices are shown in green within the original matrix, and the all-zero rows and columns in red. ]]

A matrix is '''[[full-rank]]''' when its [[Wikipedia:Rank (linear algebra)|rank]] equals whichever is smaller between its width (column count) and height (row count):

* For a ''wide'' matrix (height is smaller), it is full-rank when its rank equals its ''height''.

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===Why row-rank always equals column-rank===

[[File:Row-rank is col-rank.png|left|thumb|600x600px|Here we see an example matrix 𝐴 expressed as the product of matrices 𝑋 and 𝑌. These two matrices' shared dimension, 𝑟, is both the row-rank and the column-rank of 𝐴. In the middle and bottom rows of this diagram, we see how 𝐴 can be described in two different ways: in the middle row we see each of its rows as a linear combination of the 𝑟 rows of 𝑌, and in the bottom row we see each of its columns as a linear combination of the 𝑟 columns of 𝑋. So whether we reduce it row-style (HNF) or column-style (CHNF), we find the rank to be 𝑟. ]]

Any <math>(m,n)</math>-shaped matrix <math>A</math> can be expressed as the product of a <math>(m,r)</math>-shaped matrix <math>X</math> and a <math>(r,n)</math>-shaped matrix <math>Y</math>, such that the <math>r</math>-dimensions cancel out in the middle and we're left with an <math>(m,n)</math>-shaped matrix.

@@ Line 70: / Line 70: @@
 ==Rank-deficiency and full-rank==
+[[File:Full-rank - updated.png|thumb|600x600px|Rank of a matrix can be revealed by reducing a matrix such as with Hermite normal form (row or column style). The shapes of the reduced matrices are shown in green within the original matrix, and the all-zero rows and columns in red. ]]
 A matrix is '''[[full-rank]]''' when its [[Wikipedia:Rank (linear algebra)|rank]] equals whichever is smaller between its width (column count) and height (row count):
 * For a ''wide'' matrix (height is smaller), it is full-rank when its rank equals its ''height''.
@@ Line 107: / Line 107: @@
 ===Why row-rank always equals column-rank===
+[[File:Row-rank is col-rank.png|left|thumb|600x600px|Here we see an example matrix 𝐴 expressed as the product of matrices 𝑋 and 𝑌. These two matrices' shared dimension, 𝑟, is both the row-rank and the column-rank of 𝐴. In the middle and bottom rows of this diagram, we see how 𝐴 can be described in two different ways: in the middle row we see each of its rows as a linear combination of the 𝑟 rows of 𝑌, and in the bottom row we see each of its columns as a linear combination of the 𝑟 columns of 𝑋. So whether we reduce it row-style (HNF) or column-style (CHNF), we find the rank to be 𝑟. ]]
 Any <math>(m,n)</math>-shaped matrix <math>A</math> can be expressed as the product of a <math>(m,r)</math>-shaped matrix <math>X</math> and a <math>(r,n)</math>-shaped matrix <math>Y</math>, such that the <math>r</math>-dimensions cancel out in the middle and we're left with an <math>(m,n)</math>-shaped matrix.