Find the difference between the number of rows and columns of a matrix.
Let $B$ be an $n$ by $r$ matrix and let $M$ be the block matrix: begin{pmatrix} 0 &B\ B^t & 0 end{pmatrix}
$M$ has nullity 1. Show that the difference between the number of rows and columns of $B$ is 1.
I am supposed to use the fact that row rank = column rank for matrices over R, however I have no idea where to start.
linear-algebra linear-transformations
asked Jan 4 at 21:17
John CoxJohn Cox
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Let $B$ be an $n$ by $r$ matrix and let $M$ be the block matrix: begin{pmatrix} 0 &B\ B^t & 0 end{pmatrix}
$M$ has nullity 1. Show that the difference between the number of rows and columns of $B$ is 1.
I am supposed to use the fact that row rank = column rank for matrices over R, however I have no idea where to start.
linear-algebra linear-transformations
asked Jan 4 at 21:17
John CoxJohn Cox
13
New contributor
John Cox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
How are the nullities of $B$ and $B^t$ related?
– user3482749
Jan 4 at 21:23
Since their ranks are equal then Null(B)= Null(B^t)+r-n
– John Cox
Jan 5 at 10:19
add a comment |
Let $B$ be an $n$ by $r$ matrix and let $M$ be the block matrix: begin{pmatrix} 0 &B\ B^t & 0 end{pmatrix}
$M$ has nullity 1. Show that the difference between the number of rows and columns of $B$ is 1.
I am supposed to use the fact that row rank = column rank for matrices over R, however I have no idea where to start.
linear-algebra linear-transformations
asked Jan 4 at 21:17
John CoxJohn Cox
13
New contributor
John Cox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Let $B$ be an $n$ by $r$ matrix and let $M$ be the block matrix: begin{pmatrix} 0 &B\ B^t & 0 end{pmatrix}
$M$ has nullity 1. Show that the difference between the number of rows and columns of $B$ is 1.
I am supposed to use the fact that row rank = column rank for matrices over R, however I have no idea where to start.
linear-algebra linear-transformations
linear-algebra linear-transformations
asked Jan 4 at 21:17
John CoxJohn Cox
13
New contributor
John Cox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 21:17
John CoxJohn Cox
13
New contributor
John Cox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 21:17
John CoxJohn Cox
13
New contributor
John Cox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 21:17
John CoxJohn Cox
13
asked Jan 4 at 21:17
John CoxJohn Cox
13
13
New contributor
John Cox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
John Cox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
John Cox is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
How are the nullities of $B$ and $B^t$ related?
– user3482749
Jan 4 at 21:23
Since their ranks are equal then Null(B)= Null(B^t)+r-n
– John Cox
Jan 5 at 10:19
add a comment |
2
How are the nullities of $B$ and $B^t$ related?
– user3482749
Jan 4 at 21:23
Since their ranks are equal then Null(B)= Null(B^t)+r-n
– John Cox
Jan 5 at 10:19
2
2
How are the nullities of $B$ and $B^t$ related?
– user3482749
Jan 4 at 21:23
How are the nullities of $B$ and $B^t$ related?
– user3482749
Jan 4 at 21:23
Since their ranks are equal then Null(B)= Null(B^t)+r-n
– John Cox
Jan 5 at 10:19
Since their ranks are equal then Null(B)= Null(B^t)+r-n
– John Cox
Jan 5 at 10:19
add a comment |
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2
How are the nullities of $B$ and $B^t$ related?
– user3482749
Jan 4 at 21:23
Since their ranks are equal then Null(B)= Null(B^t)+r-n
– John Cox
Jan 5 at 10:19