If $A$ is a matrix $10$ by $12$ and $A x=b$ is solvable for every $b$, then th column space of $A$ is?
$begingroup$
If $A$ is a matrix $10$ by $12$ and $A x=b$ is solvable for every $b$, then th column space of $A$ is?
The column space of $A$ is the whole of $R^m$ which is $12$ is this answer the right answer ?
linear-algebra matrices matrix-rank
$endgroup$
add a comment |
$begingroup$
If $A$ is a matrix $10$ by $12$ and $A x=b$ is solvable for every $b$, then th column space of $A$ is?
The column space of $A$ is the whole of $R^m$ which is $12$ is this answer the right answer ?
linear-algebra matrices matrix-rank
$endgroup$
add a comment |
$begingroup$
If $A$ is a matrix $10$ by $12$ and $A x=b$ is solvable for every $b$, then th column space of $A$ is?
The column space of $A$ is the whole of $R^m$ which is $12$ is this answer the right answer ?
linear-algebra matrices matrix-rank
$endgroup$
If $A$ is a matrix $10$ by $12$ and $A x=b$ is solvable for every $b$, then th column space of $A$ is?
The column space of $A$ is the whole of $R^m$ which is $12$ is this answer the right answer ?
linear-algebra matrices matrix-rank
linear-algebra matrices matrix-rank
edited Dec 12 '18 at 13:19
MathOverview
8,78243163
8,78243163
asked Dec 11 '18 at 20:52
The BeardThe Beard
54
54
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
According to the hypothesis of your problem, for all fixed vector
$
b=begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$
in $mathbb{R}^{10}$ exists a vector
$
x=begin{pmatrix} x_1 \vdots \ x_j\ vdots \x_{12} end{pmatrix}
$
in $mathbb{R}^{12}$ such that the equality below is true
$$
begin{pmatrix}
A_{11} & ldots & A_{1j} & ldots & A_{1,12}
\
vdots & ddots & vdots & ddots & vdots
\
A_{i1} & ldots & A_{ij} & ldots & A_{i,12}
\
vdots & ddots & vdots & ddots & vdots
\
A_{10,1} & ldots & A_{10,j} & ldots & A_{10,12}
end{pmatrix}
begin{pmatrix} x_1 \vdots \ x_j\ vdots \x_{12} end{pmatrix}
=
begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$$
By performing the multiplication of the matrix $A$ by the column matrix $x$ we have that the following statement is valid: for all fixed $binmathbb{R}^{10}$ there are numbers $x_1$, $ldots$, $x_j$, $ldots$, $x_{12}$ such that
$$
begin{pmatrix}
x_1cdot A_{11}+& ldots &+x_jcdot A_{1j}+ & ldots &+x_{12}cdot A_{1,12}
\
vdots & ddots & vdots & ddots & vdots
\
x_1cdot A_{i1}+ & ldots &+x_jcdot A_{ij}+ & ldots &+x_{12}cdot A_{i,12}
\
vdots & ddots & vdots & ddots & vdots
\
x_1cdot A_{10,1}+ & ldots & +x_jcdot A_{10,j} +& ldots &+x_{12}cdot A_{10,12}
end{pmatrix}
=
begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$$
Let the vector $A_j$ be the j-th column of matrix $A$.The last statement can then be rewritten as follows. For all vector $binmathbb{R}^{10}$ there are real numbers $x_1$, $ldots$, $x_j$, $ldots$, $x_{12}$ such that
$$
x_1 begin{pmatrix} A_{11} \vdots \ A_{i1}\ vdots \A_{10 ,1} end{pmatrix}
+ldots +
x_j begin{pmatrix} A_{1j} \vdots \ A_{ij}\ vdots \A_{10 ,j} end{pmatrix}
+ldots +
x_{12} begin{pmatrix} A_{1,12} \vdots \ A_{i,12}\ vdots \A_{10 ,12} end{pmatrix}
=
begin{pmatrix} b_{1} \vdots \ b_{j}\ vdots \b_{10} end{pmatrix}
$$
Since the vector $ b $ is arbitrary, it follows that the columns $A_1,ldots, A_{12} $ of $ A $ are vectors that generate the space $mathbb {R}^{10} $.
$endgroup$
$begingroup$
thank you very much for this great explanation so the column space of A are the vectors that generate the space R^10
$endgroup$
– The Beard
Dec 12 '18 at 14:38
add a comment |
$begingroup$
If $A$ is a $10 times 12$ matrix then $A$ is the matrix associated with a linear transformation $T:mathbb{R}^{12}tomathbb{R}^{10}$. Since $Ax=b$ is solvable so it has full rank.
Hence, Column Space ($A) = $ Range($T$) = $mathbb{R}^{10}$.
$endgroup$
$begingroup$
thanks you for the answer although iam bit confused as it is conflict with the first answer
$endgroup$
– The Beard
Dec 11 '18 at 21:22
$begingroup$
@TheBeard: See if we go by the rules of matrix multiplication, we have $A$, which is $10times12$ matrix, is multiplied with $x$ which is $12times 1$ matrix giving $b$ which should be a $10times 1 $ matrix or $bin mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:28
$begingroup$
@TheBeard: Column space is the span of the columns of the matrix $A$ which are vectors in $mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:33
$begingroup$
thank you very much i understand it now
$endgroup$
– The Beard
Dec 11 '18 at 21:56
add a comment |
$begingroup$
Your answer is not correct. The maximum rank of $A$ can be only $10$. Since $AX=b$ has a solution for every $b$, this means the rank of $A$ must be $10$. Thus two ($12-10=2$) columns must be free. This means the column space is $Bbb{R}^{10}$.
$endgroup$
1
$begingroup$
thank you for explaining that to me
$endgroup$
– The Beard
Dec 11 '18 at 21:10
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
According to the hypothesis of your problem, for all fixed vector
$
b=begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$
in $mathbb{R}^{10}$ exists a vector
$
x=begin{pmatrix} x_1 \vdots \ x_j\ vdots \x_{12} end{pmatrix}
$
in $mathbb{R}^{12}$ such that the equality below is true
$$
begin{pmatrix}
A_{11} & ldots & A_{1j} & ldots & A_{1,12}
\
vdots & ddots & vdots & ddots & vdots
\
A_{i1} & ldots & A_{ij} & ldots & A_{i,12}
\
vdots & ddots & vdots & ddots & vdots
\
A_{10,1} & ldots & A_{10,j} & ldots & A_{10,12}
end{pmatrix}
begin{pmatrix} x_1 \vdots \ x_j\ vdots \x_{12} end{pmatrix}
=
begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$$
By performing the multiplication of the matrix $A$ by the column matrix $x$ we have that the following statement is valid: for all fixed $binmathbb{R}^{10}$ there are numbers $x_1$, $ldots$, $x_j$, $ldots$, $x_{12}$ such that
$$
begin{pmatrix}
x_1cdot A_{11}+& ldots &+x_jcdot A_{1j}+ & ldots &+x_{12}cdot A_{1,12}
\
vdots & ddots & vdots & ddots & vdots
\
x_1cdot A_{i1}+ & ldots &+x_jcdot A_{ij}+ & ldots &+x_{12}cdot A_{i,12}
\
vdots & ddots & vdots & ddots & vdots
\
x_1cdot A_{10,1}+ & ldots & +x_jcdot A_{10,j} +& ldots &+x_{12}cdot A_{10,12}
end{pmatrix}
=
begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$$
Let the vector $A_j$ be the j-th column of matrix $A$.The last statement can then be rewritten as follows. For all vector $binmathbb{R}^{10}$ there are real numbers $x_1$, $ldots$, $x_j$, $ldots$, $x_{12}$ such that
$$
x_1 begin{pmatrix} A_{11} \vdots \ A_{i1}\ vdots \A_{10 ,1} end{pmatrix}
+ldots +
x_j begin{pmatrix} A_{1j} \vdots \ A_{ij}\ vdots \A_{10 ,j} end{pmatrix}
+ldots +
x_{12} begin{pmatrix} A_{1,12} \vdots \ A_{i,12}\ vdots \A_{10 ,12} end{pmatrix}
=
begin{pmatrix} b_{1} \vdots \ b_{j}\ vdots \b_{10} end{pmatrix}
$$
Since the vector $ b $ is arbitrary, it follows that the columns $A_1,ldots, A_{12} $ of $ A $ are vectors that generate the space $mathbb {R}^{10} $.
$endgroup$
$begingroup$
thank you very much for this great explanation so the column space of A are the vectors that generate the space R^10
$endgroup$
– The Beard
Dec 12 '18 at 14:38
add a comment |
$begingroup$
According to the hypothesis of your problem, for all fixed vector
$
b=begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$
in $mathbb{R}^{10}$ exists a vector
$
x=begin{pmatrix} x_1 \vdots \ x_j\ vdots \x_{12} end{pmatrix}
$
in $mathbb{R}^{12}$ such that the equality below is true
$$
begin{pmatrix}
A_{11} & ldots & A_{1j} & ldots & A_{1,12}
\
vdots & ddots & vdots & ddots & vdots
\
A_{i1} & ldots & A_{ij} & ldots & A_{i,12}
\
vdots & ddots & vdots & ddots & vdots
\
A_{10,1} & ldots & A_{10,j} & ldots & A_{10,12}
end{pmatrix}
begin{pmatrix} x_1 \vdots \ x_j\ vdots \x_{12} end{pmatrix}
=
begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$$
By performing the multiplication of the matrix $A$ by the column matrix $x$ we have that the following statement is valid: for all fixed $binmathbb{R}^{10}$ there are numbers $x_1$, $ldots$, $x_j$, $ldots$, $x_{12}$ such that
$$
begin{pmatrix}
x_1cdot A_{11}+& ldots &+x_jcdot A_{1j}+ & ldots &+x_{12}cdot A_{1,12}
\
vdots & ddots & vdots & ddots & vdots
\
x_1cdot A_{i1}+ & ldots &+x_jcdot A_{ij}+ & ldots &+x_{12}cdot A_{i,12}
\
vdots & ddots & vdots & ddots & vdots
\
x_1cdot A_{10,1}+ & ldots & +x_jcdot A_{10,j} +& ldots &+x_{12}cdot A_{10,12}
end{pmatrix}
=
begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$$
Let the vector $A_j$ be the j-th column of matrix $A$.The last statement can then be rewritten as follows. For all vector $binmathbb{R}^{10}$ there are real numbers $x_1$, $ldots$, $x_j$, $ldots$, $x_{12}$ such that
$$
x_1 begin{pmatrix} A_{11} \vdots \ A_{i1}\ vdots \A_{10 ,1} end{pmatrix}
+ldots +
x_j begin{pmatrix} A_{1j} \vdots \ A_{ij}\ vdots \A_{10 ,j} end{pmatrix}
+ldots +
x_{12} begin{pmatrix} A_{1,12} \vdots \ A_{i,12}\ vdots \A_{10 ,12} end{pmatrix}
=
begin{pmatrix} b_{1} \vdots \ b_{j}\ vdots \b_{10} end{pmatrix}
$$
Since the vector $ b $ is arbitrary, it follows that the columns $A_1,ldots, A_{12} $ of $ A $ are vectors that generate the space $mathbb {R}^{10} $.
$endgroup$
$begingroup$
thank you very much for this great explanation so the column space of A are the vectors that generate the space R^10
$endgroup$
– The Beard
Dec 12 '18 at 14:38
add a comment |
$begingroup$
According to the hypothesis of your problem, for all fixed vector
$
b=begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$
in $mathbb{R}^{10}$ exists a vector
$
x=begin{pmatrix} x_1 \vdots \ x_j\ vdots \x_{12} end{pmatrix}
$
in $mathbb{R}^{12}$ such that the equality below is true
$$
begin{pmatrix}
A_{11} & ldots & A_{1j} & ldots & A_{1,12}
\
vdots & ddots & vdots & ddots & vdots
\
A_{i1} & ldots & A_{ij} & ldots & A_{i,12}
\
vdots & ddots & vdots & ddots & vdots
\
A_{10,1} & ldots & A_{10,j} & ldots & A_{10,12}
end{pmatrix}
begin{pmatrix} x_1 \vdots \ x_j\ vdots \x_{12} end{pmatrix}
=
begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$$
By performing the multiplication of the matrix $A$ by the column matrix $x$ we have that the following statement is valid: for all fixed $binmathbb{R}^{10}$ there are numbers $x_1$, $ldots$, $x_j$, $ldots$, $x_{12}$ such that
$$
begin{pmatrix}
x_1cdot A_{11}+& ldots &+x_jcdot A_{1j}+ & ldots &+x_{12}cdot A_{1,12}
\
vdots & ddots & vdots & ddots & vdots
\
x_1cdot A_{i1}+ & ldots &+x_jcdot A_{ij}+ & ldots &+x_{12}cdot A_{i,12}
\
vdots & ddots & vdots & ddots & vdots
\
x_1cdot A_{10,1}+ & ldots & +x_jcdot A_{10,j} +& ldots &+x_{12}cdot A_{10,12}
end{pmatrix}
=
begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$$
Let the vector $A_j$ be the j-th column of matrix $A$.The last statement can then be rewritten as follows. For all vector $binmathbb{R}^{10}$ there are real numbers $x_1$, $ldots$, $x_j$, $ldots$, $x_{12}$ such that
$$
x_1 begin{pmatrix} A_{11} \vdots \ A_{i1}\ vdots \A_{10 ,1} end{pmatrix}
+ldots +
x_j begin{pmatrix} A_{1j} \vdots \ A_{ij}\ vdots \A_{10 ,j} end{pmatrix}
+ldots +
x_{12} begin{pmatrix} A_{1,12} \vdots \ A_{i,12}\ vdots \A_{10 ,12} end{pmatrix}
=
begin{pmatrix} b_{1} \vdots \ b_{j}\ vdots \b_{10} end{pmatrix}
$$
Since the vector $ b $ is arbitrary, it follows that the columns $A_1,ldots, A_{12} $ of $ A $ are vectors that generate the space $mathbb {R}^{10} $.
$endgroup$
According to the hypothesis of your problem, for all fixed vector
$
b=begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$
in $mathbb{R}^{10}$ exists a vector
$
x=begin{pmatrix} x_1 \vdots \ x_j\ vdots \x_{12} end{pmatrix}
$
in $mathbb{R}^{12}$ such that the equality below is true
$$
begin{pmatrix}
A_{11} & ldots & A_{1j} & ldots & A_{1,12}
\
vdots & ddots & vdots & ddots & vdots
\
A_{i1} & ldots & A_{ij} & ldots & A_{i,12}
\
vdots & ddots & vdots & ddots & vdots
\
A_{10,1} & ldots & A_{10,j} & ldots & A_{10,12}
end{pmatrix}
begin{pmatrix} x_1 \vdots \ x_j\ vdots \x_{12} end{pmatrix}
=
begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$$
By performing the multiplication of the matrix $A$ by the column matrix $x$ we have that the following statement is valid: for all fixed $binmathbb{R}^{10}$ there are numbers $x_1$, $ldots$, $x_j$, $ldots$, $x_{12}$ such that
$$
begin{pmatrix}
x_1cdot A_{11}+& ldots &+x_jcdot A_{1j}+ & ldots &+x_{12}cdot A_{1,12}
\
vdots & ddots & vdots & ddots & vdots
\
x_1cdot A_{i1}+ & ldots &+x_jcdot A_{ij}+ & ldots &+x_{12}cdot A_{i,12}
\
vdots & ddots & vdots & ddots & vdots
\
x_1cdot A_{10,1}+ & ldots & +x_jcdot A_{10,j} +& ldots &+x_{12}cdot A_{10,12}
end{pmatrix}
=
begin{pmatrix} b_1 \vdots \ b_j\ vdots \b_{10} end{pmatrix}
$$
Let the vector $A_j$ be the j-th column of matrix $A$.The last statement can then be rewritten as follows. For all vector $binmathbb{R}^{10}$ there are real numbers $x_1$, $ldots$, $x_j$, $ldots$, $x_{12}$ such that
$$
x_1 begin{pmatrix} A_{11} \vdots \ A_{i1}\ vdots \A_{10 ,1} end{pmatrix}
+ldots +
x_j begin{pmatrix} A_{1j} \vdots \ A_{ij}\ vdots \A_{10 ,j} end{pmatrix}
+ldots +
x_{12} begin{pmatrix} A_{1,12} \vdots \ A_{i,12}\ vdots \A_{10 ,12} end{pmatrix}
=
begin{pmatrix} b_{1} \vdots \ b_{j}\ vdots \b_{10} end{pmatrix}
$$
Since the vector $ b $ is arbitrary, it follows that the columns $A_1,ldots, A_{12} $ of $ A $ are vectors that generate the space $mathbb {R}^{10} $.
edited Dec 23 '18 at 16:33
answered Dec 12 '18 at 14:22
MathOverviewMathOverview
8,78243163
8,78243163
$begingroup$
thank you very much for this great explanation so the column space of A are the vectors that generate the space R^10
$endgroup$
– The Beard
Dec 12 '18 at 14:38
add a comment |
$begingroup$
thank you very much for this great explanation so the column space of A are the vectors that generate the space R^10
$endgroup$
– The Beard
Dec 12 '18 at 14:38
$begingroup$
thank you very much for this great explanation so the column space of A are the vectors that generate the space R^10
$endgroup$
– The Beard
Dec 12 '18 at 14:38
$begingroup$
thank you very much for this great explanation so the column space of A are the vectors that generate the space R^10
$endgroup$
– The Beard
Dec 12 '18 at 14:38
add a comment |
$begingroup$
If $A$ is a $10 times 12$ matrix then $A$ is the matrix associated with a linear transformation $T:mathbb{R}^{12}tomathbb{R}^{10}$. Since $Ax=b$ is solvable so it has full rank.
Hence, Column Space ($A) = $ Range($T$) = $mathbb{R}^{10}$.
$endgroup$
$begingroup$
thanks you for the answer although iam bit confused as it is conflict with the first answer
$endgroup$
– The Beard
Dec 11 '18 at 21:22
$begingroup$
@TheBeard: See if we go by the rules of matrix multiplication, we have $A$, which is $10times12$ matrix, is multiplied with $x$ which is $12times 1$ matrix giving $b$ which should be a $10times 1 $ matrix or $bin mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:28
$begingroup$
@TheBeard: Column space is the span of the columns of the matrix $A$ which are vectors in $mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:33
$begingroup$
thank you very much i understand it now
$endgroup$
– The Beard
Dec 11 '18 at 21:56
add a comment |
$begingroup$
If $A$ is a $10 times 12$ matrix then $A$ is the matrix associated with a linear transformation $T:mathbb{R}^{12}tomathbb{R}^{10}$. Since $Ax=b$ is solvable so it has full rank.
Hence, Column Space ($A) = $ Range($T$) = $mathbb{R}^{10}$.
$endgroup$
$begingroup$
thanks you for the answer although iam bit confused as it is conflict with the first answer
$endgroup$
– The Beard
Dec 11 '18 at 21:22
$begingroup$
@TheBeard: See if we go by the rules of matrix multiplication, we have $A$, which is $10times12$ matrix, is multiplied with $x$ which is $12times 1$ matrix giving $b$ which should be a $10times 1 $ matrix or $bin mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:28
$begingroup$
@TheBeard: Column space is the span of the columns of the matrix $A$ which are vectors in $mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:33
$begingroup$
thank you very much i understand it now
$endgroup$
– The Beard
Dec 11 '18 at 21:56
add a comment |
$begingroup$
If $A$ is a $10 times 12$ matrix then $A$ is the matrix associated with a linear transformation $T:mathbb{R}^{12}tomathbb{R}^{10}$. Since $Ax=b$ is solvable so it has full rank.
Hence, Column Space ($A) = $ Range($T$) = $mathbb{R}^{10}$.
$endgroup$
If $A$ is a $10 times 12$ matrix then $A$ is the matrix associated with a linear transformation $T:mathbb{R}^{12}tomathbb{R}^{10}$. Since $Ax=b$ is solvable so it has full rank.
Hence, Column Space ($A) = $ Range($T$) = $mathbb{R}^{10}$.
answered Dec 11 '18 at 21:13
Yadati KiranYadati Kiran
1,7911619
1,7911619
$begingroup$
thanks you for the answer although iam bit confused as it is conflict with the first answer
$endgroup$
– The Beard
Dec 11 '18 at 21:22
$begingroup$
@TheBeard: See if we go by the rules of matrix multiplication, we have $A$, which is $10times12$ matrix, is multiplied with $x$ which is $12times 1$ matrix giving $b$ which should be a $10times 1 $ matrix or $bin mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:28
$begingroup$
@TheBeard: Column space is the span of the columns of the matrix $A$ which are vectors in $mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:33
$begingroup$
thank you very much i understand it now
$endgroup$
– The Beard
Dec 11 '18 at 21:56
add a comment |
$begingroup$
thanks you for the answer although iam bit confused as it is conflict with the first answer
$endgroup$
– The Beard
Dec 11 '18 at 21:22
$begingroup$
@TheBeard: See if we go by the rules of matrix multiplication, we have $A$, which is $10times12$ matrix, is multiplied with $x$ which is $12times 1$ matrix giving $b$ which should be a $10times 1 $ matrix or $bin mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:28
$begingroup$
@TheBeard: Column space is the span of the columns of the matrix $A$ which are vectors in $mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:33
$begingroup$
thank you very much i understand it now
$endgroup$
– The Beard
Dec 11 '18 at 21:56
$begingroup$
thanks you for the answer although iam bit confused as it is conflict with the first answer
$endgroup$
– The Beard
Dec 11 '18 at 21:22
$begingroup$
thanks you for the answer although iam bit confused as it is conflict with the first answer
$endgroup$
– The Beard
Dec 11 '18 at 21:22
$begingroup$
@TheBeard: See if we go by the rules of matrix multiplication, we have $A$, which is $10times12$ matrix, is multiplied with $x$ which is $12times 1$ matrix giving $b$ which should be a $10times 1 $ matrix or $bin mathbb{R}^{10}$.
$endgroup$
– Yadati Kiran
Dec 11 '18 at 21:28
$begingroup$
@TheBeard: See if we go by the rules of matrix multiplication, we have $A$, which is $10times12$ matrix, is multiplied with $x$ which is $12times 1$ matrix giving $b$ which should be a $10times 1 $ matrix or $bin mathbb{R}^{10}$.
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– Yadati Kiran
Dec 11 '18 at 21:28
$begingroup$
@TheBeard: Column space is the span of the columns of the matrix $A$ which are vectors in $mathbb{R}^{10}$.
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– Yadati Kiran
Dec 11 '18 at 21:33
$begingroup$
@TheBeard: Column space is the span of the columns of the matrix $A$ which are vectors in $mathbb{R}^{10}$.
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– Yadati Kiran
Dec 11 '18 at 21:33
$begingroup$
thank you very much i understand it now
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– The Beard
Dec 11 '18 at 21:56
$begingroup$
thank you very much i understand it now
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– The Beard
Dec 11 '18 at 21:56
add a comment |
$begingroup$
Your answer is not correct. The maximum rank of $A$ can be only $10$. Since $AX=b$ has a solution for every $b$, this means the rank of $A$ must be $10$. Thus two ($12-10=2$) columns must be free. This means the column space is $Bbb{R}^{10}$.
$endgroup$
1
$begingroup$
thank you for explaining that to me
$endgroup$
– The Beard
Dec 11 '18 at 21:10
add a comment |
$begingroup$
Your answer is not correct. The maximum rank of $A$ can be only $10$. Since $AX=b$ has a solution for every $b$, this means the rank of $A$ must be $10$. Thus two ($12-10=2$) columns must be free. This means the column space is $Bbb{R}^{10}$.
$endgroup$
1
$begingroup$
thank you for explaining that to me
$endgroup$
– The Beard
Dec 11 '18 at 21:10
add a comment |
$begingroup$
Your answer is not correct. The maximum rank of $A$ can be only $10$. Since $AX=b$ has a solution for every $b$, this means the rank of $A$ must be $10$. Thus two ($12-10=2$) columns must be free. This means the column space is $Bbb{R}^{10}$.
$endgroup$
Your answer is not correct. The maximum rank of $A$ can be only $10$. Since $AX=b$ has a solution for every $b$, this means the rank of $A$ must be $10$. Thus two ($12-10=2$) columns must be free. This means the column space is $Bbb{R}^{10}$.
edited Dec 11 '18 at 21:33
answered Dec 11 '18 at 20:59
Anurag AAnurag A
26.2k12251
26.2k12251
1
$begingroup$
thank you for explaining that to me
$endgroup$
– The Beard
Dec 11 '18 at 21:10
add a comment |
1
$begingroup$
thank you for explaining that to me
$endgroup$
– The Beard
Dec 11 '18 at 21:10
1
1
$begingroup$
thank you for explaining that to me
$endgroup$
– The Beard
Dec 11 '18 at 21:10
$begingroup$
thank you for explaining that to me
$endgroup$
– The Beard
Dec 11 '18 at 21:10
add a comment |
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