Mixing
Element of transformation range verses element of column space
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I was wondering if the question "Is $b$ in the range of the linear transformation $x mapsto Ax$?" is synonymous to asking "Is $b in ColumnSpace(A)$?".
linear-algebra matrices
asked Jan 29 at 20:16
kylemartkylemart
1465
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add a comment |
$begingroup$
I was wondering if the question "Is $b$ in the range of the linear transformation $x mapsto Ax$?" is synonymous to asking "Is $b in ColumnSpace(A)$?".
linear-algebra matrices
asked Jan 29 at 20:16
kylemartkylemart
1465
$endgroup$
add a comment |
$begingroup$
I was wondering if the question "Is $b$ in the range of the linear transformation $x mapsto Ax$?" is synonymous to asking "Is $b in ColumnSpace(A)$?".
linear-algebra matrices
asked Jan 29 at 20:16
kylemartkylemart
1465
$endgroup$
I was wondering if the question "Is $b$ in the range of the linear transformation $x mapsto Ax$?" is synonymous to asking "Is $b in ColumnSpace(A)$?".
linear-algebra matrices
linear-algebra matrices
asked Jan 29 at 20:16
kylemartkylemart
1465
asked Jan 29 at 20:16
kylemartkylemart
1465
asked Jan 29 at 20:16
kylemartkylemart
1465
asked Jan 29 at 20:16
kylemartkylemart
1465
asked Jan 29 at 20:16
kylemartkylemart
1465
1465
add a comment |
add a comment |
1 Answer
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$begingroup$
Yes, this is generally synonymous: The column space of a matrix $A$ is (by definition) the span of its column vectors, i.e. it contains elements of the form $lambda_{1}c_{1} + ... + lambda_{n}c_{n}$ where the $c_{i}$ are the columns of the matrix. If $b in ColumnSpace(A)$ this means b is expressible in this form. But this is exactly the same as $b$ being in the image of your linear transformation. (You can see this by multiplying your matrix $A$ with a general vector).
answered Jan 29 at 20:50
IamalgebraicIamalgebraic
364
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yes, this is generally synonymous: The column space of a matrix $A$ is (by definition) the span of its column vectors, i.e. it contains elements of the form $lambda_{1}c_{1} + ... + lambda_{n}c_{n}$ where the $c_{i}$ are the columns of the matrix. If $b in ColumnSpace(A)$ this means b is expressible in this form. But this is exactly the same as $b$ being in the image of your linear transformation. (You can see this by multiplying your matrix $A$ with a general vector).
answered Jan 29 at 20:50
IamalgebraicIamalgebraic
364
$endgroup$
add a comment |
$begingroup$
Yes, this is generally synonymous: The column space of a matrix $A$ is (by definition) the span of its column vectors, i.e. it contains elements of the form $lambda_{1}c_{1} + ... + lambda_{n}c_{n}$ where the $c_{i}$ are the columns of the matrix. If $b in ColumnSpace(A)$ this means b is expressible in this form. But this is exactly the same as $b$ being in the image of your linear transformation. (You can see this by multiplying your matrix $A$ with a general vector).
answered Jan 29 at 20:50
IamalgebraicIamalgebraic
364
$endgroup$
add a comment |
$begingroup$
Yes, this is generally synonymous: The column space of a matrix $A$ is (by definition) the span of its column vectors, i.e. it contains elements of the form $lambda_{1}c_{1} + ... + lambda_{n}c_{n}$ where the $c_{i}$ are the columns of the matrix. If $b in ColumnSpace(A)$ this means b is expressible in this form. But this is exactly the same as $b$ being in the image of your linear transformation. (You can see this by multiplying your matrix $A$ with a general vector).
answered Jan 29 at 20:50
IamalgebraicIamalgebraic
364
$endgroup$
Yes, this is generally synonymous: The column space of a matrix $A$ is (by definition) the span of its column vectors, i.e. it contains elements of the form $lambda_{1}c_{1} + ... + lambda_{n}c_{n}$ where the $c_{i}$ are the columns of the matrix. If $b in ColumnSpace(A)$ this means b is expressible in this form. But this is exactly the same as $b$ being in the image of your linear transformation. (You can see this by multiplying your matrix $A$ with a general vector).
answered Jan 29 at 20:50
IamalgebraicIamalgebraic
364
answered Jan 29 at 20:50
IamalgebraicIamalgebraic
364
answered Jan 29 at 20:50
IamalgebraicIamalgebraic
364
answered Jan 29 at 20:50
IamalgebraicIamalgebraic
364
364
add a comment |
add a comment |
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