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How come that if Ax = W is consistent than a vector w is part of the Im A.
Determining the value of $h$ that makes a linear system consistent.How to show that a given set is a vector space?Can I use determinants to show that two vector sets span the same subspace?If the equation $Ax = b$ is consitent, can the equation have more than one solution for some $b$?Differentiating the Norm of a Matrix Vector Producthow to show a general consistent nonhomogeneous system (if that is even needed)?How come that HSL can contain more information than RGB?Prove that the maximal number of linearly independent vectors for the given vectors in the $k$ -vector space $k^3$ is 0 or 2.The linear system $Bx = b$ is consistent $iff$ $BB^Tx = b$ is consistentDetermine for which values of 𝑘 the following system is consistent
$begingroup$
If I have a matrix A and a vector w and I want to verify that w is included in the Im A, Why is solving [ A | w ] and it giving a solution (being consistent) prove that w is include in Im A? I am also not sure what consistent even means.
linear-algebra
asked Mar 29 at 2:53
Dr.StoneDr.Stone
335
$endgroup$
add a comment |
$begingroup$
If I have a matrix A and a vector w and I want to verify that w is included in the Im A, Why is solving [ A | w ] and it giving a solution (being consistent) prove that w is include in Im A? I am also not sure what consistent even means.
linear-algebra
asked Mar 29 at 2:53
Dr.StoneDr.Stone
335
$endgroup$
$begingroup$
How do you define the image/column space of a matrix? Also, consistency of a linear system of equations is covered in this Wikipedia article.
$endgroup$
– Brian
Mar 29 at 2:56
$begingroup$
"$Ax = w$ is consistent" is defined as "$Ax = w$ has a solution", i.e., "There is a vector $x$ such that $Ax = w$", that is, $w in operatornameIm A$.
$endgroup$
– M. Vinay
Mar 29 at 2:56
add a comment |
$begingroup$
If I have a matrix A and a vector w and I want to verify that w is included in the Im A, Why is solving [ A | w ] and it giving a solution (being consistent) prove that w is include in Im A? I am also not sure what consistent even means.
linear-algebra
asked Mar 29 at 2:53
Dr.StoneDr.Stone
335
$endgroup$
If I have a matrix A and a vector w and I want to verify that w is included in the Im A, Why is solving [ A | w ] and it giving a solution (being consistent) prove that w is include in Im A? I am also not sure what consistent even means.
linear-algebra
linear-algebra
asked Mar 29 at 2:53
Dr.StoneDr.Stone
335
asked Mar 29 at 2:53
Dr.StoneDr.Stone
335
asked Mar 29 at 2:53
Dr.StoneDr.Stone
335
asked Mar 29 at 2:53
Dr.StoneDr.Stone
335
asked Mar 29 at 2:53
Dr.StoneDr.Stone
335
335
$begingroup$
How do you define the image/column space of a matrix? Also, consistency of a linear system of equations is covered in this Wikipedia article.
$endgroup$
– Brian
Mar 29 at 2:56
$begingroup$
"$Ax = w$ is consistent" is defined as "$Ax = w$ has a solution", i.e., "There is a vector $x$ such that $Ax = w$", that is, $w in operatornameIm A$.
$endgroup$
– M. Vinay
Mar 29 at 2:56
add a comment |
$begingroup$
How do you define the image/column space of a matrix? Also, consistency of a linear system of equations is covered in this Wikipedia article.
$endgroup$
– Brian
Mar 29 at 2:56
$begingroup$
"$Ax = w$ is consistent" is defined as "$Ax = w$ has a solution", i.e., "There is a vector $x$ such that $Ax = w$", that is, $w in operatornameIm A$.
$endgroup$
– M. Vinay
Mar 29 at 2:56
$begingroup$
How do you define the image/column space of a matrix? Also, consistency of a linear system of equations is covered in this Wikipedia article.
$endgroup$
– Brian
Mar 29 at 2:56
$begingroup$
How do you define the image/column space of a matrix? Also, consistency of a linear system of equations is covered in this Wikipedia article.
$endgroup$
– Brian
Mar 29 at 2:56
$begingroup$
"$Ax = w$ is consistent" is defined as "$Ax = w$ has a solution", i.e., "There is a vector $x$ such that $Ax = w$", that is, $w in operatornameIm A$.
$endgroup$
– M. Vinay
Mar 29 at 2:56
$begingroup$
"$Ax = w$ is consistent" is defined as "$Ax = w$ has a solution", i.e., "There is a vector $x$ such that $Ax = w$", that is, $w in operatornameIm A$.
$endgroup$
– M. Vinay
Mar 29 at 2:56
add a comment |
1 Answer
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$begingroup$
A matrix equation $Ax = w$ is consistent if it has at least one solution $x$. If any solutions $x$ exist, then $w$ is in the image, or range of the matrix $A$. Think of the matrix $A$ as a function that maps a vector $x$ to another vector $w$. The output $w$ must be in the image of $A$.
answered Mar 29 at 2:57
D.B.D.B.
1,49019
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$begingroup$
A matrix equation $Ax = w$ is consistent if it has at least one solution $x$. If any solutions $x$ exist, then $w$ is in the image, or range of the matrix $A$. Think of the matrix $A$ as a function that maps a vector $x$ to another vector $w$. The output $w$ must be in the image of $A$.
answered Mar 29 at 2:57
D.B.D.B.
1,49019
$endgroup$
add a comment |
$begingroup$
A matrix equation $Ax = w$ is consistent if it has at least one solution $x$. If any solutions $x$ exist, then $w$ is in the image, or range of the matrix $A$. Think of the matrix $A$ as a function that maps a vector $x$ to another vector $w$. The output $w$ must be in the image of $A$.
answered Mar 29 at 2:57
D.B.D.B.
1,49019
$endgroup$
add a comment |
$begingroup$
A matrix equation $Ax = w$ is consistent if it has at least one solution $x$. If any solutions $x$ exist, then $w$ is in the image, or range of the matrix $A$. Think of the matrix $A$ as a function that maps a vector $x$ to another vector $w$. The output $w$ must be in the image of $A$.
answered Mar 29 at 2:57
D.B.D.B.
1,49019
$endgroup$
A matrix equation $Ax = w$ is consistent if it has at least one solution $x$. If any solutions $x$ exist, then $w$ is in the image, or range of the matrix $A$. Think of the matrix $A$ as a function that maps a vector $x$ to another vector $w$. The output $w$ must be in the image of $A$.
answered Mar 29 at 2:57
D.B.D.B.
1,49019
answered Mar 29 at 2:57
D.B.D.B.
1,49019
answered Mar 29 at 2:57
D.B.D.B.
1,49019
answered Mar 29 at 2:57
D.B.D.B.
1,49019
1,49019
add a comment |
add a comment |
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$begingroup$
How do you define the image/column space of a matrix? Also, consistency of a linear system of equations is covered in this Wikipedia article.
$endgroup$
– Brian
Mar 29 at 2:56
$begingroup$
"$Ax = w$ is consistent" is defined as "$Ax = w$ has a solution", i.e., "There is a vector $x$ such that $Ax = w$", that is, $w in operatornameIm A$.
$endgroup$
– M. Vinay
Mar 29 at 2:56