Why is the change of basis formula counter-intuitive? [See details] Announcing the arrival of...
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Why is the change of basis formula counter-intuitive? [See details]
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Change of basisChange of basis = similarity?Change of Basis vs. Linear TransformationMatrices for change of basis linear transformationsConfusion about change of basis matrixIntuitive understanding of the $BAB^{-1}$ formula for changing basis in linear transformations.Standard Basis and Change of Basis MatrixStandard matrix linear transformation - change of basisHard change of basis/ linear transformation problemChange of basis difference between linear and bilinear transformation
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The formula of change of basis $[T]_{B'} = P_{B' <-B}[T]_{B}P_{B <- B'}$.
I don't understand why you need $P_{B <- B'}$? It seems to me that if you have the transformation expressed in B already with $[T]_{B}$ you just need to translate to B' by using $P_{B' <-B}$ to get $[T]_{B'}$ rendering $P_{B <- B'}$ as useless. Can someone explain what I am missing here?
linear-algebra
asked 4 hours ago
Dr.StoneDr.Stone
626
$endgroup$
add a comment |
$begingroup$
The formula of change of basis $[T]_{B'} = P_{B' <-B}[T]_{B}P_{B <- B'}$.
I don't understand why you need $P_{B <- B'}$? It seems to me that if you have the transformation expressed in B already with $[T]_{B}$ you just need to translate to B' by using $P_{B' <-B}$ to get $[T]_{B'}$ rendering $P_{B <- B'}$ as useless. Can someone explain what I am missing here?
linear-algebra
asked 4 hours ago
Dr.StoneDr.Stone
626
$endgroup$
$begingroup$
@littleO this is actually what I was looking for. Can you write it as real answer instead of a comment it might help others understand as well so I can approve it.
$endgroup$
– Dr.Stone
4 hours ago
add a comment |
$begingroup$
The formula of change of basis $[T]_{B'} = P_{B' <-B}[T]_{B}P_{B <- B'}$.
I don't understand why you need $P_{B <- B'}$? It seems to me that if you have the transformation expressed in B already with $[T]_{B}$ you just need to translate to B' by using $P_{B' <-B}$ to get $[T]_{B'}$ rendering $P_{B <- B'}$ as useless. Can someone explain what I am missing here?
linear-algebra
asked 4 hours ago
Dr.StoneDr.Stone
626
$endgroup$
The formula of change of basis $[T]_{B'} = P_{B' <-B}[T]_{B}P_{B <- B'}$.
I don't understand why you need $P_{B <- B'}$? It seems to me that if you have the transformation expressed in B already with $[T]_{B}$ you just need to translate to B' by using $P_{B' <-B}$ to get $[T]_{B'}$ rendering $P_{B <- B'}$ as useless. Can someone explain what I am missing here?
linear-algebra
linear-algebra
asked 4 hours ago
Dr.StoneDr.Stone
626
asked 4 hours ago
Dr.StoneDr.Stone
626
asked 4 hours ago
Dr.StoneDr.Stone
626
asked 4 hours ago
Dr.StoneDr.Stone
626
asked 4 hours ago
Dr.StoneDr.Stone
626
626
$begingroup$
@littleO this is actually what I was looking for. Can you write it as real answer instead of a comment it might help others understand as well so I can approve it.
$endgroup$
– Dr.Stone
4 hours ago
add a comment |
$begingroup$
@littleO this is actually what I was looking for. Can you write it as real answer instead of a comment it might help others understand as well so I can approve it.
$endgroup$
– Dr.Stone
4 hours ago
$begingroup$
@littleO this is actually what I was looking for. Can you write it as real answer instead of a comment it might help others understand as well so I can approve it.
$endgroup$
– Dr.Stone
4 hours ago
$begingroup$
@littleO this is actually what I was looking for. Can you write it as real answer instead of a comment it might help others understand as well so I can approve it.
$endgroup$
– Dr.Stone
4 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Imagine what you must do to a vector expressed in $B'$ coordinates in order to apply $T$ to it. First you switch from $B'$ coordinates to $B$ coordinates, then you multiply by the matrix of $T$ (with respect to $B$), then finally you switch back to $B'$ coordinates.
answered 4 hours ago
littleOlittleO
30.6k649111
$endgroup$
add a comment |
$begingroup$
Write $B={e_1,...,e_n}, B' ={e_1',...,e_n'}$
If you have the first member of $B'$, $e_1'$, and you want to compute the effect of $T$ on it, then applying $[T]_B$ to $(1,0,...0)$ will be the effect of $T$ on the first member of the basis $B$, so $e_1$, written in the basis $B$ so it has nothing to do with the image of $e_1'$.
So if you only know $[T]_B$ and want to compute $Te_1'$, then you first have to write $e_1'$ in the basis $B$, so you compute $P_{B'to B}(1,0,...0)$, then compute $[T]_B$ times that, which yields $Te_1'$ but written in the basis $B$, so now you have to write it in the basis $B'$ to get the correct result, that's where $P_{Bto B'}$ comes from on the left. This gives the formula
answered 4 hours ago
MaxMax
16.6k11144
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Imagine what you must do to a vector expressed in $B'$ coordinates in order to apply $T$ to it. First you switch from $B'$ coordinates to $B$ coordinates, then you multiply by the matrix of $T$ (with respect to $B$), then finally you switch back to $B'$ coordinates.
answered 4 hours ago
littleOlittleO
30.6k649111
$endgroup$
add a comment |
$begingroup$
Imagine what you must do to a vector expressed in $B'$ coordinates in order to apply $T$ to it. First you switch from $B'$ coordinates to $B$ coordinates, then you multiply by the matrix of $T$ (with respect to $B$), then finally you switch back to $B'$ coordinates.
answered 4 hours ago
littleOlittleO
30.6k649111
$endgroup$
add a comment |
$begingroup$
Imagine what you must do to a vector expressed in $B'$ coordinates in order to apply $T$ to it. First you switch from $B'$ coordinates to $B$ coordinates, then you multiply by the matrix of $T$ (with respect to $B$), then finally you switch back to $B'$ coordinates.
answered 4 hours ago
littleOlittleO
30.6k649111
$endgroup$
Imagine what you must do to a vector expressed in $B'$ coordinates in order to apply $T$ to it. First you switch from $B'$ coordinates to $B$ coordinates, then you multiply by the matrix of $T$ (with respect to $B$), then finally you switch back to $B'$ coordinates.
answered 4 hours ago
littleOlittleO
30.6k649111
answered 4 hours ago
littleOlittleO
30.6k649111
answered 4 hours ago
littleOlittleO
30.6k649111
answered 4 hours ago
littleOlittleO
30.6k649111
30.6k649111
add a comment |
add a comment |
$begingroup$
Write $B={e_1,...,e_n}, B' ={e_1',...,e_n'}$
If you have the first member of $B'$, $e_1'$, and you want to compute the effect of $T$ on it, then applying $[T]_B$ to $(1,0,...0)$ will be the effect of $T$ on the first member of the basis $B$, so $e_1$, written in the basis $B$ so it has nothing to do with the image of $e_1'$.
So if you only know $[T]_B$ and want to compute $Te_1'$, then you first have to write $e_1'$ in the basis $B$, so you compute $P_{B'to B}(1,0,...0)$, then compute $[T]_B$ times that, which yields $Te_1'$ but written in the basis $B$, so now you have to write it in the basis $B'$ to get the correct result, that's where $P_{Bto B'}$ comes from on the left. This gives the formula
answered 4 hours ago
MaxMax
16.6k11144
$endgroup$
add a comment |
$begingroup$
Write $B={e_1,...,e_n}, B' ={e_1',...,e_n'}$
If you have the first member of $B'$, $e_1'$, and you want to compute the effect of $T$ on it, then applying $[T]_B$ to $(1,0,...0)$ will be the effect of $T$ on the first member of the basis $B$, so $e_1$, written in the basis $B$ so it has nothing to do with the image of $e_1'$.
So if you only know $[T]_B$ and want to compute $Te_1'$, then you first have to write $e_1'$ in the basis $B$, so you compute $P_{B'to B}(1,0,...0)$, then compute $[T]_B$ times that, which yields $Te_1'$ but written in the basis $B$, so now you have to write it in the basis $B'$ to get the correct result, that's where $P_{Bto B'}$ comes from on the left. This gives the formula
answered 4 hours ago
MaxMax
16.6k11144
$endgroup$
add a comment |
$begingroup$
Write $B={e_1,...,e_n}, B' ={e_1',...,e_n'}$
If you have the first member of $B'$, $e_1'$, and you want to compute the effect of $T$ on it, then applying $[T]_B$ to $(1,0,...0)$ will be the effect of $T$ on the first member of the basis $B$, so $e_1$, written in the basis $B$ so it has nothing to do with the image of $e_1'$.
So if you only know $[T]_B$ and want to compute $Te_1'$, then you first have to write $e_1'$ in the basis $B$, so you compute $P_{B'to B}(1,0,...0)$, then compute $[T]_B$ times that, which yields $Te_1'$ but written in the basis $B$, so now you have to write it in the basis $B'$ to get the correct result, that's where $P_{Bto B'}$ comes from on the left. This gives the formula
answered 4 hours ago
MaxMax
16.6k11144
$endgroup$
Write $B={e_1,...,e_n}, B' ={e_1',...,e_n'}$
If you have the first member of $B'$, $e_1'$, and you want to compute the effect of $T$ on it, then applying $[T]_B$ to $(1,0,...0)$ will be the effect of $T$ on the first member of the basis $B$, so $e_1$, written in the basis $B$ so it has nothing to do with the image of $e_1'$.
So if you only know $[T]_B$ and want to compute $Te_1'$, then you first have to write $e_1'$ in the basis $B$, so you compute $P_{B'to B}(1,0,...0)$, then compute $[T]_B$ times that, which yields $Te_1'$ but written in the basis $B$, so now you have to write it in the basis $B'$ to get the correct result, that's where $P_{Bto B'}$ comes from on the left. This gives the formula
answered 4 hours ago
MaxMax
16.6k11144
answered 4 hours ago
MaxMax
16.6k11144
answered 4 hours ago
MaxMax
16.6k11144
answered 4 hours ago
MaxMax
16.6k11144
16.6k11144
add a comment |
add a comment |
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$begingroup$
@littleO this is actually what I was looking for. Can you write it as real answer instead of a comment it might help others understand as well so I can approve it.
$endgroup$
– Dr.Stone
4 hours ago