question about determinant variety in Karen Smith's “An Invitation to Algebraic Geometry”

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I don't understand in the example below:

If $k geq n$, then the determinant variety is the whole space ?

linear-algebra algebraic-geometry

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edited Jul 22 at 16:38

asked Jul 22 at 16:20

Newbie

384111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
  – darij grinberg
  Jul 22 at 18:15
  

  
  
  
  

add a comment | 

up vote
2
down vote

favorite

1

I don't understand in the example below:

If $k geq n$, then the determinant variety is the whole space ?

linear-algebra algebraic-geometry

share|cite|improve this question

edited Jul 22 at 16:38

asked Jul 22 at 16:20

Newbie

384111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
  – darij grinberg
  Jul 22 at 18:15
  

  
  
  
  

add a comment | 

up vote
2
down vote

favorite

1

up vote
2
down vote

favorite

1

1

I don't understand in the example below:

If $k geq n$, then the determinant variety is the whole space ?

linear-algebra algebraic-geometry

share|cite|improve this question

edited Jul 22 at 16:38

asked Jul 22 at 16:20

Newbie

384111

I don't understand in the example below:

If $k geq n$, then the determinant variety is the whole space ?

linear-algebra algebraic-geometry

share|cite|improve this question

edited Jul 22 at 16:38

asked Jul 22 at 16:20

Newbie

384111

share|cite|improve this question

share|cite|improve this question

edited Jul 22 at 16:38

edited Jul 22 at 16:38

edited Jul 22 at 16:38

asked Jul 22 at 16:20

Newbie

384111

asked Jul 22 at 16:20

Newbie

384111

asked Jul 22 at 16:20

Newbie

384111

384111

• 
  
  
  

  
  

  
  
  
  
  
  

  
  She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
  – darij grinberg
  Jul 22 at 18:15
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
  – darij grinberg
  Jul 22 at 18:15
  

  
  
  
  

She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
– darij grinberg
Jul 22 at 18:15

She is talking about $n times n$-matrices. If $k geq n$, then each $n times n$-matrix has rank at most $k$.
– darij grinberg
Jul 22 at 18:15

add a comment | 

2 Answers
2

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4
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accepted

An $n times n$ matrix has $n$ rows and $n$ columns. The rank of a matrix is the number of linearly independent rows or columns (row rank and column rank being the same). The greatest possible number of linearly independent vectors out of a set of $n$ vectors is obviously $n$. Thus for $k ge n$, every $n times n$ matrix $A$ satisfies

$textrank(A) le n le k. tag 1$

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answered Jul 22 at 16:31

Robert Lewis

36.9k22255

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up vote
3
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Every $n times n$ matrix has rank at most $n$.

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answered Jul 22 at 16:26

Hurkyl

108k9112253

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
4
down vote



accepted

An $n times n$ matrix has $n$ rows and $n$ columns. The rank of a matrix is the number of linearly independent rows or columns (row rank and column rank being the same). The greatest possible number of linearly independent vectors out of a set of $n$ vectors is obviously $n$. Thus for $k ge n$, every $n times n$ matrix $A$ satisfies

$textrank(A) le n le k. tag 1$

share|cite|improve this answer

answered Jul 22 at 16:31

Robert Lewis

36.9k22255

add a comment | 

up vote
4
down vote



accepted

An $n times n$ matrix has $n$ rows and $n$ columns. The rank of a matrix is the number of linearly independent rows or columns (row rank and column rank being the same). The greatest possible number of linearly independent vectors out of a set of $n$ vectors is obviously $n$. Thus for $k ge n$, every $n times n$ matrix $A$ satisfies

$textrank(A) le n le k. tag 1$

share|cite|improve this answer

answered Jul 22 at 16:31

Robert Lewis

36.9k22255

add a comment | 

up vote
4
down vote



accepted

up vote
4
down vote



accepted

An $n times n$ matrix has $n$ rows and $n$ columns. The rank of a matrix is the number of linearly independent rows or columns (row rank and column rank being the same). The greatest possible number of linearly independent vectors out of a set of $n$ vectors is obviously $n$. Thus for $k ge n$, every $n times n$ matrix $A$ satisfies

$textrank(A) le n le k. tag 1$

share|cite|improve this answer

answered Jul 22 at 16:31

Robert Lewis

36.9k22255

An $n times n$ matrix has $n$ rows and $n$ columns. The rank of a matrix is the number of linearly independent rows or columns (row rank and column rank being the same). The greatest possible number of linearly independent vectors out of a set of $n$ vectors is obviously $n$. Thus for $k ge n$, every $n times n$ matrix $A$ satisfies

$textrank(A) le n le k. tag 1$

share|cite|improve this answer

answered Jul 22 at 16:31

Robert Lewis

36.9k22255

share|cite|improve this answer

share|cite|improve this answer

answered Jul 22 at 16:31

Robert Lewis

36.9k22255

answered Jul 22 at 16:31

Robert Lewis

36.9k22255

answered Jul 22 at 16:31

Robert Lewis

36.9k22255

36.9k22255

add a comment | 

add a comment | 

up vote
3
down vote

Every $n times n$ matrix has rank at most $n$.

share|cite|improve this answer

answered Jul 22 at 16:26

Hurkyl

108k9112253

add a comment | 

up vote
3
down vote

Every $n times n$ matrix has rank at most $n$.

share|cite|improve this answer

answered Jul 22 at 16:26

Hurkyl

108k9112253

add a comment | 

up vote
3
down vote

up vote
3
down vote

Every $n times n$ matrix has rank at most $n$.

share|cite|improve this answer

answered Jul 22 at 16:26

Hurkyl

108k9112253

Every $n times n$ matrix has rank at most $n$.

share|cite|improve this answer

answered Jul 22 at 16:26

Hurkyl

108k9112253

share|cite|improve this answer

share|cite|improve this answer

answered Jul 22 at 16:26

Hurkyl

108k9112253

answered Jul 22 at 16:26

Hurkyl

108k9112253

answered Jul 22 at 16:26

Hurkyl

108k9112253

108k9112253

add a comment | 

add a comment | 

 

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