Rank of a Linear Operator over matrix space [on hold]

Clash Royale CLAN TAG#URR8PPP

up vote
0
down vote

favorite

Suppose $Qin M_10times 10(mathbbR)$ is a matrix of rank 5. Let $T:M_10times 10(mathbbR)to M_10times 10(mathbbR)$ be the linear
transformation defined by $T(P)=QP$. Then the rank of $T$ is?

linear-algebra linear-transformations

share|cite|improve this question

asked Aug 3 at 9:34

user90533

212112

put on hold as off-topic by uniquesolution, user477343, Strants, John Ma, José Carlos Santos Aug 3 at 18:25

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – uniquesolution, user477343, Strants, John Ma, José Carlos Santos

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  @uniquesolution Just to be aware, you might want to head here. I mean, I believe your question is by far appropriate, but try not to ask it too many times.
  – user477343
  Aug 3 at 10:16
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @user477343 - Thank you for your enlightening comment, which makes clear what you have tried.
  – uniquesolution
  Aug 3 at 10:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @uniquesolution I have no idea what a matrix space is, nor what a linear transformation is... what does $M_10times 10$ mean as a function? I don't know.
  – user477343
  Aug 3 at 10:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $M_10times 10(mathbbR)$ means the set of all $10times 10$ matrices with real coefficients. If you don't know what a linear transformation is, I suggest you go back to your textbook because it is one of the first things you learn in a LInear Algebra Course.
  – daruma
  Aug 3 at 10:34
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  depends on $P$, say if $P in mathbbM_10 times k(mathbfR)$ is of rank 10 for some $k$, then $textrank(QP) = 5$.
  – pointguard0
  Aug 3 at 11:31
  
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

Suppose $Qin M_10times 10(mathbbR)$ is a matrix of rank 5. Let $T:M_10times 10(mathbbR)to M_10times 10(mathbbR)$ be the linear
transformation defined by $T(P)=QP$. Then the rank of $T$ is?

linear-algebra linear-transformations

share|cite|improve this question

asked Aug 3 at 9:34

user90533

212112

put on hold as off-topic by uniquesolution, user477343, Strants, John Ma, José Carlos Santos Aug 3 at 18:25

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – uniquesolution, user477343, Strants, John Ma, José Carlos Santos

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  @uniquesolution Just to be aware, you might want to head here. I mean, I believe your question is by far appropriate, but try not to ask it too many times.
  – user477343
  Aug 3 at 10:16
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @user477343 - Thank you for your enlightening comment, which makes clear what you have tried.
  – uniquesolution
  Aug 3 at 10:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @uniquesolution I have no idea what a matrix space is, nor what a linear transformation is... what does $M_10times 10$ mean as a function? I don't know.
  – user477343
  Aug 3 at 10:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $M_10times 10(mathbbR)$ means the set of all $10times 10$ matrices with real coefficients. If you don't know what a linear transformation is, I suggest you go back to your textbook because it is one of the first things you learn in a LInear Algebra Course.
  – daruma
  Aug 3 at 10:34
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  depends on $P$, say if $P in mathbbM_10 times k(mathbfR)$ is of rank 10 for some $k$, then $textrank(QP) = 5$.
  – pointguard0
  Aug 3 at 11:31
  
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Suppose $Qin M_10times 10(mathbbR)$ is a matrix of rank 5. Let $T:M_10times 10(mathbbR)to M_10times 10(mathbbR)$ be the linear
transformation defined by $T(P)=QP$. Then the rank of $T$ is?

linear-algebra linear-transformations

share|cite|improve this question

asked Aug 3 at 9:34

user90533

212112

Suppose $Qin M_10times 10(mathbbR)$ is a matrix of rank 5. Let $T:M_10times 10(mathbbR)to M_10times 10(mathbbR)$ be the linear
transformation defined by $T(P)=QP$. Then the rank of $T$ is?

linear-algebra linear-transformations

share|cite|improve this question

asked Aug 3 at 9:34

user90533

212112

share|cite|improve this question

share|cite|improve this question

asked Aug 3 at 9:34

user90533

212112

asked Aug 3 at 9:34

user90533

212112

asked Aug 3 at 9:34

user90533

212112

212112

put on hold as off-topic by uniquesolution, user477343, Strants, John Ma, José Carlos Santos Aug 3 at 18:25

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – uniquesolution, user477343, Strants, John Ma, José Carlos Santos

If this question can be reworded to fit the rules in the help center, please edit the question.

put on hold as off-topic by uniquesolution, user477343, Strants, John Ma, José Carlos Santos Aug 3 at 18:25

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – uniquesolution, user477343, Strants, John Ma, José Carlos Santos

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  @uniquesolution Just to be aware, you might want to head here. I mean, I believe your question is by far appropriate, but try not to ask it too many times.
  – user477343
  Aug 3 at 10:16
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @user477343 - Thank you for your enlightening comment, which makes clear what you have tried.
  – uniquesolution
  Aug 3 at 10:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @uniquesolution I have no idea what a matrix space is, nor what a linear transformation is... what does $M_10times 10$ mean as a function? I don't know.
  – user477343
  Aug 3 at 10:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $M_10times 10(mathbbR)$ means the set of all $10times 10$ matrices with real coefficients. If you don't know what a linear transformation is, I suggest you go back to your textbook because it is one of the first things you learn in a LInear Algebra Course.
  – daruma
  Aug 3 at 10:34
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  depends on $P$, say if $P in mathbbM_10 times k(mathbfR)$ is of rank 10 for some $k$, then $textrank(QP) = 5$.
  – pointguard0
  Aug 3 at 11:31
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  @uniquesolution Just to be aware, you might want to head here. I mean, I believe your question is by far appropriate, but try not to ask it too many times.
  – user477343
  Aug 3 at 10:16
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  @user477343 - Thank you for your enlightening comment, which makes clear what you have tried.
  – uniquesolution
  Aug 3 at 10:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @uniquesolution I have no idea what a matrix space is, nor what a linear transformation is... what does $M_10times 10$ mean as a function? I don't know.
  – user477343
  Aug 3 at 10:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $M_10times 10(mathbbR)$ means the set of all $10times 10$ matrices with real coefficients. If you don't know what a linear transformation is, I suggest you go back to your textbook because it is one of the first things you learn in a LInear Algebra Course.
  – daruma
  Aug 3 at 10:34
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  depends on $P$, say if $P in mathbbM_10 times k(mathbfR)$ is of rank 10 for some $k$, then $textrank(QP) = 5$.
  – pointguard0
  Aug 3 at 11:31
  
  

  
  
  
  

@uniquesolution Just to be aware, you might want to head here. I mean, I believe your question is by far appropriate, but try not to ask it too many times.
– user477343
Aug 3 at 10:16

@uniquesolution Just to be aware, you might want to head here. I mean, I believe your question is by far appropriate, but try not to ask it too many times.
– user477343
Aug 3 at 10:16

1

1

@user477343 - Thank you for your enlightening comment, which makes clear what you have tried.
– uniquesolution
Aug 3 at 10:24

@user477343 - Thank you for your enlightening comment, which makes clear what you have tried.
– uniquesolution
Aug 3 at 10:24

@uniquesolution I have no idea what a matrix space is, nor what a linear transformation is... what does $M_10times 10$ mean as a function? I don't know.
– user477343
Aug 3 at 10:27

@uniquesolution I have no idea what a matrix space is, nor what a linear transformation is... what does $M_10times 10$ mean as a function? I don't know.
– user477343
Aug 3 at 10:27

$M_10times 10(mathbbR)$ means the set of all $10times 10$ matrices with real coefficients. If you don't know what a linear transformation is, I suggest you go back to your textbook because it is one of the first things you learn in a LInear Algebra Course.
– daruma
Aug 3 at 10:34

$M_10times 10(mathbbR)$ means the set of all $10times 10$ matrices with real coefficients. If you don't know what a linear transformation is, I suggest you go back to your textbook because it is one of the first things you learn in a LInear Algebra Course.
– daruma
Aug 3 at 10:34

depends on $P$, say if $P in mathbbM_10 times k(mathbfR)$ is of rank 10 for some $k$, then $textrank(QP) = 5$.
– pointguard0
Aug 3 at 11:31

depends on $P$, say if $P in mathbbM_10 times k(mathbfR)$ is of rank 10 for some $k$, then $textrank(QP) = 5$.
– pointguard0
Aug 3 at 11:31

add a comment | 

1 Answer
1

active

oldest

votes

up vote
1
down vote

Let $m$ and $n$ be positive integers with $mleq n$. Fix a base field $mathbbK$ and write $mathcalR:=textMat_ntimes n(mathbbK)$. Suppose that $Qin mathcalR$ is of rank $m$. We shall prove that the linear transformation $T_Q:Rto R$ sending $Pmapsto QP$ for all $Pin mathcalR$ is a linear map of rank $mn$. That is, we shall prove that $dim_mathbbKbig(textim(T)big)=mn$.

Let $q_k$ be the $k$-th column vector of $Q$ for $kin 1,2,ldots,n=:[n]$. Suppose that $q_j_1,q_j_2,ldots,q_j_m$ are linearly independent columns of $Q$ where $j_1,j_2,ldots,j_min[n]$ satisfies $j_1<j_2<ldots<j_m$. Write $E_i,j$ for the matrix with $1$ at the $(i,j)$-entry and $0$ everywhere else (where $i,jin [n]$). Then, the image $X_i,j:=T_Q(E_i,j)$ of $E_i,j$ under $T_Q$ is the matrix where the $j$-th column equals $q_i$, and other columns are $0$. It is easy to prove that the matrices $X_i,j$ for $iinj_1,j_2,ldots,j_m$ and $jin[n]$ are linearly independent elements in $mathcalR$ which span $textim(T)$, and I shall leave this task to the OP.

share|cite|improve this answer

answered Aug 3 at 12:21

Batominovski

22.6k22776

add a comment | 

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote

Let $m$ and $n$ be positive integers with $mleq n$. Fix a base field $mathbbK$ and write $mathcalR:=textMat_ntimes n(mathbbK)$. Suppose that $Qin mathcalR$ is of rank $m$. We shall prove that the linear transformation $T_Q:Rto R$ sending $Pmapsto QP$ for all $Pin mathcalR$ is a linear map of rank $mn$. That is, we shall prove that $dim_mathbbKbig(textim(T)big)=mn$.

Let $q_k$ be the $k$-th column vector of $Q$ for $kin 1,2,ldots,n=:[n]$. Suppose that $q_j_1,q_j_2,ldots,q_j_m$ are linearly independent columns of $Q$ where $j_1,j_2,ldots,j_min[n]$ satisfies $j_1<j_2<ldots<j_m$. Write $E_i,j$ for the matrix with $1$ at the $(i,j)$-entry and $0$ everywhere else (where $i,jin [n]$). Then, the image $X_i,j:=T_Q(E_i,j)$ of $E_i,j$ under $T_Q$ is the matrix where the $j$-th column equals $q_i$, and other columns are $0$. It is easy to prove that the matrices $X_i,j$ for $iinj_1,j_2,ldots,j_m$ and $jin[n]$ are linearly independent elements in $mathcalR$ which span $textim(T)$, and I shall leave this task to the OP.

share|cite|improve this answer

answered Aug 3 at 12:21

Batominovski

22.6k22776

add a comment | 

up vote
1
down vote

Let $m$ and $n$ be positive integers with $mleq n$. Fix a base field $mathbbK$ and write $mathcalR:=textMat_ntimes n(mathbbK)$. Suppose that $Qin mathcalR$ is of rank $m$. We shall prove that the linear transformation $T_Q:Rto R$ sending $Pmapsto QP$ for all $Pin mathcalR$ is a linear map of rank $mn$. That is, we shall prove that $dim_mathbbKbig(textim(T)big)=mn$.

Let $q_k$ be the $k$-th column vector of $Q$ for $kin 1,2,ldots,n=:[n]$. Suppose that $q_j_1,q_j_2,ldots,q_j_m$ are linearly independent columns of $Q$ where $j_1,j_2,ldots,j_min[n]$ satisfies $j_1<j_2<ldots<j_m$. Write $E_i,j$ for the matrix with $1$ at the $(i,j)$-entry and $0$ everywhere else (where $i,jin [n]$). Then, the image $X_i,j:=T_Q(E_i,j)$ of $E_i,j$ under $T_Q$ is the matrix where the $j$-th column equals $q_i$, and other columns are $0$. It is easy to prove that the matrices $X_i,j$ for $iinj_1,j_2,ldots,j_m$ and $jin[n]$ are linearly independent elements in $mathcalR$ which span $textim(T)$, and I shall leave this task to the OP.

share|cite|improve this answer

answered Aug 3 at 12:21

Batominovski

22.6k22776

add a comment | 

up vote
1
down vote

up vote
1
down vote

Let $m$ and $n$ be positive integers with $mleq n$. Fix a base field $mathbbK$ and write $mathcalR:=textMat_ntimes n(mathbbK)$. Suppose that $Qin mathcalR$ is of rank $m$. We shall prove that the linear transformation $T_Q:Rto R$ sending $Pmapsto QP$ for all $Pin mathcalR$ is a linear map of rank $mn$. That is, we shall prove that $dim_mathbbKbig(textim(T)big)=mn$.

Let $q_k$ be the $k$-th column vector of $Q$ for $kin 1,2,ldots,n=:[n]$. Suppose that $q_j_1,q_j_2,ldots,q_j_m$ are linearly independent columns of $Q$ where $j_1,j_2,ldots,j_min[n]$ satisfies $j_1<j_2<ldots<j_m$. Write $E_i,j$ for the matrix with $1$ at the $(i,j)$-entry and $0$ everywhere else (where $i,jin [n]$). Then, the image $X_i,j:=T_Q(E_i,j)$ of $E_i,j$ under $T_Q$ is the matrix where the $j$-th column equals $q_i$, and other columns are $0$. It is easy to prove that the matrices $X_i,j$ for $iinj_1,j_2,ldots,j_m$ and $jin[n]$ are linearly independent elements in $mathcalR$ which span $textim(T)$, and I shall leave this task to the OP.

share|cite|improve this answer

answered Aug 3 at 12:21

Batominovski

22.6k22776

Let $m$ and $n$ be positive integers with $mleq n$. Fix a base field $mathbbK$ and write $mathcalR:=textMat_ntimes n(mathbbK)$. Suppose that $Qin mathcalR$ is of rank $m$. We shall prove that the linear transformation $T_Q:Rto R$ sending $Pmapsto QP$ for all $Pin mathcalR$ is a linear map of rank $mn$. That is, we shall prove that $dim_mathbbKbig(textim(T)big)=mn$.

Let $q_k$ be the $k$-th column vector of $Q$ for $kin 1,2,ldots,n=:[n]$. Suppose that $q_j_1,q_j_2,ldots,q_j_m$ are linearly independent columns of $Q$ where $j_1,j_2,ldots,j_min[n]$ satisfies $j_1<j_2<ldots<j_m$. Write $E_i,j$ for the matrix with $1$ at the $(i,j)$-entry and $0$ everywhere else (where $i,jin [n]$). Then, the image $X_i,j:=T_Q(E_i,j)$ of $E_i,j$ under $T_Q$ is the matrix where the $j$-th column equals $q_i$, and other columns are $0$. It is easy to prove that the matrices $X_i,j$ for $iinj_1,j_2,ldots,j_m$ and $jin[n]$ are linearly independent elements in $mathcalR$ which span $textim(T)$, and I shall leave this task to the OP.

share|cite|improve this answer

answered Aug 3 at 12:21

Batominovski

22.6k22776

share|cite|improve this answer

share|cite|improve this answer

answered Aug 3 at 12:21

Batominovski

22.6k22776

answered Aug 3 at 12:21

Batominovski

22.6k22776

answered Aug 3 at 12:21

Batominovski

22.6k22776

22.6k22776

add a comment | 

add a comment |