Given any $beta in mathbbR^2$ can I find an orthogonal matrix $K$ such that $Kbeta= vert beta vert beginbmatrix 1 \ 0endbmatrix$?If $A$ is orthogonal, for any vector $x$ such that $Ax = b$, $Vert x Vert = Vert b Vert$Given matrix $B =beginbmatrix0&-1\0&0endbmatrix$, find $e^Bt$ from a short infinite seriesGiven an eigen values evaluate $S*tinybeginbmatrix 0\1\0 endbmatrix$Find a formula for $Tleft(beginbmatrix x_1 \ x_2 endbmatrixright)$$T colon mathbbR^2 rightarrow mathbbR$ such that $Tbeginbmatrix 1 \1 endbmatrix=3$ and $Tbeginbmatrix -1 \2 endbmatrix=6$$M= beginbmatrix 1 & -1 & 1\ 2 & 1 & 2 endbmatrix$,find a rank one $2 times 3$ matrix $N$ such that $|M-N|_2$ is minimizedrank $beginbmatrix-asin(t)\acos(t)\betaendbmatrix=1$?find a matrix $C in mathbbF_2$ such thatFind matrix $A$ of $Ax=b$ if $b=beginbmatrix 1 \ 2 \ 1 \ endbmatrix \$When is a matrix similar to $beginbmatrix 0 & 1 \ 0 & 0 \ endbmatrix$
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Given any $beta in mathbbR^2$ can I find an orthogonal matrix $K$ such that $Kbeta= vert beta vert beginbmatrix 1 \ 0endbmatrix$?
If $A$ is orthogonal, for any vector $x$ such that $Ax = b$, $Vert x Vert = Vert b Vert$Given matrix $B =beginbmatrix0&-1\0&0endbmatrix$, find $e^Bt$ from a short infinite seriesGiven an eigen values evaluate $S*tinybeginbmatrix 0\1\0 endbmatrix$Find a formula for $Tleft(beginbmatrix x_1 \ x_2 endbmatrixright)$$T colon mathbbR^2 rightarrow mathbbR$ such that $Tbeginbmatrix 1 \1 endbmatrix=3$ and $Tbeginbmatrix -1 \2 endbmatrix=6$$M= beginbmatrix 1 & -1 & 1\ 2 & 1 & 2 endbmatrix$,find a rank one $2 times 3$ matrix $N$ such that $|M-N|_2$ is minimizedrank $beginbmatrix-asin(t)\acos(t)\betaendbmatrix=1$?find a matrix $C in mathbbF_2$ such thatFind matrix $A$ of $Ax=b$ if $b=beginbmatrix 1 \ 2 \ 1 \ endbmatrix \$When is a matrix similar to $beginbmatrix 0 & 1 \ 0 & 0 \ endbmatrix$
$begingroup$
Given any $beta in mathbbR^2$(not the zero vector) can I find an orthogonal matrix $K$ such that $Kbeta= vert beta vert beginbmatrix 1 \ 0endbmatrix$.
I am not sure how do I actually start reasoning about this problem. Even hints would be appreciated
linear-algebra
asked Mar 17 at 21:01
user3503589user3503589
1,3101821
$endgroup$
add a comment |
$begingroup$
Given any $beta in mathbbR^2$(not the zero vector) can I find an orthogonal matrix $K$ such that $Kbeta= vert beta vert beginbmatrix 1 \ 0endbmatrix$.
I am not sure how do I actually start reasoning about this problem. Even hints would be appreciated
linear-algebra
asked Mar 17 at 21:01
user3503589user3503589
1,3101821
$endgroup$
2
$begingroup$
hints- try to do it for special values of $beta$, remind yourself what an orthogonal matrix represents geometrically
$endgroup$
– Calvin Khor
Mar 17 at 21:16
$begingroup$
I tried to show it for $beta =(c,0)'$ and $beta=(0,c)$ and I can find orthogonal matrices in these case. Can I ask you if its true or if I am looking for a counter example
$endgroup$
– user3503589
Mar 17 at 21:28
1
$begingroup$
So we can think of $K$ as a matrix which rotates $beta$ until it is on the $x$-axis?
$endgroup$
– user3503589
Mar 17 at 21:33
$begingroup$
Its true, and yes!
$endgroup$
– Calvin Khor
Mar 17 at 21:50
add a comment |
$begingroup$
Given any $beta in mathbbR^2$(not the zero vector) can I find an orthogonal matrix $K$ such that $Kbeta= vert beta vert beginbmatrix 1 \ 0endbmatrix$.
I am not sure how do I actually start reasoning about this problem. Even hints would be appreciated
linear-algebra
asked Mar 17 at 21:01
user3503589user3503589
1,3101821
$endgroup$
Given any $beta in mathbbR^2$(not the zero vector) can I find an orthogonal matrix $K$ such that $Kbeta= vert beta vert beginbmatrix 1 \ 0endbmatrix$.
I am not sure how do I actually start reasoning about this problem. Even hints would be appreciated
linear-algebra
linear-algebra
asked Mar 17 at 21:01
user3503589user3503589
1,3101821
asked Mar 17 at 21:01
user3503589user3503589
1,3101821
asked Mar 17 at 21:01
user3503589user3503589
1,3101821
asked Mar 17 at 21:01
user3503589user3503589
1,3101821
asked Mar 17 at 21:01
user3503589user3503589
1,3101821
1,3101821
2
$begingroup$
hints- try to do it for special values of $beta$, remind yourself what an orthogonal matrix represents geometrically
$endgroup$
– Calvin Khor
Mar 17 at 21:16
$begingroup$
I tried to show it for $beta =(c,0)'$ and $beta=(0,c)$ and I can find orthogonal matrices in these case. Can I ask you if its true or if I am looking for a counter example
$endgroup$
– user3503589
Mar 17 at 21:28
1
$begingroup$
So we can think of $K$ as a matrix which rotates $beta$ until it is on the $x$-axis?
$endgroup$
– user3503589
Mar 17 at 21:33
$begingroup$
Its true, and yes!
$endgroup$
– Calvin Khor
Mar 17 at 21:50
add a comment |
2
$begingroup$
hints- try to do it for special values of $beta$, remind yourself what an orthogonal matrix represents geometrically
$endgroup$
– Calvin Khor
Mar 17 at 21:16
$begingroup$
I tried to show it for $beta =(c,0)'$ and $beta=(0,c)$ and I can find orthogonal matrices in these case. Can I ask you if its true or if I am looking for a counter example
$endgroup$
– user3503589
Mar 17 at 21:28
1
$begingroup$
So we can think of $K$ as a matrix which rotates $beta$ until it is on the $x$-axis?
$endgroup$
– user3503589
Mar 17 at 21:33
$begingroup$
Its true, and yes!
$endgroup$
– Calvin Khor
Mar 17 at 21:50
2
2
$begingroup$
hints- try to do it for special values of $beta$, remind yourself what an orthogonal matrix represents geometrically
$endgroup$
– Calvin Khor
Mar 17 at 21:16
$begingroup$
hints- try to do it for special values of $beta$, remind yourself what an orthogonal matrix represents geometrically
$endgroup$
– Calvin Khor
Mar 17 at 21:16
$begingroup$
I tried to show it for $beta =(c,0)'$ and $beta=(0,c)$ and I can find orthogonal matrices in these case. Can I ask you if its true or if I am looking for a counter example
$endgroup$
– user3503589
Mar 17 at 21:28
$begingroup$
I tried to show it for $beta =(c,0)'$ and $beta=(0,c)$ and I can find orthogonal matrices in these case. Can I ask you if its true or if I am looking for a counter example
$endgroup$
– user3503589
Mar 17 at 21:28
1
1
$begingroup$
So we can think of $K$ as a matrix which rotates $beta$ until it is on the $x$-axis?
$endgroup$
– user3503589
Mar 17 at 21:33
$begingroup$
So we can think of $K$ as a matrix which rotates $beta$ until it is on the $x$-axis?
$endgroup$
– user3503589
Mar 17 at 21:33
$begingroup$
Its true, and yes!
$endgroup$
– Calvin Khor
Mar 17 at 21:50
$begingroup$
Its true, and yes!
$endgroup$
– Calvin Khor
Mar 17 at 21:50
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint Look for
$$K=beginbmatrix
cos(theta) & sin(theta) \
-sin(theta) & cos(theta)
endbmatrix$$
and write $beta$ in polar coordinates
$$beta=beginbmatrix
Rcos(phi) \
Rsin(phi)
endbmatrix$$
answered Mar 17 at 21:53
N. S.N. S.
105k7114210
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
oldest
votes
$begingroup$
Hint Look for
$$K=beginbmatrix
cos(theta) & sin(theta) \
-sin(theta) & cos(theta)
endbmatrix$$
and write $beta$ in polar coordinates
$$beta=beginbmatrix
Rcos(phi) \
Rsin(phi)
endbmatrix$$
answered Mar 17 at 21:53
N. S.N. S.
105k7114210
$endgroup$
add a comment |
$begingroup$
Hint Look for
$$K=beginbmatrix
cos(theta) & sin(theta) \
-sin(theta) & cos(theta)
endbmatrix$$
and write $beta$ in polar coordinates
$$beta=beginbmatrix
Rcos(phi) \
Rsin(phi)
endbmatrix$$
answered Mar 17 at 21:53
N. S.N. S.
105k7114210
$endgroup$
add a comment |
$begingroup$
Hint Look for
$$K=beginbmatrix
cos(theta) & sin(theta) \
-sin(theta) & cos(theta)
endbmatrix$$
and write $beta$ in polar coordinates
$$beta=beginbmatrix
Rcos(phi) \
Rsin(phi)
endbmatrix$$
answered Mar 17 at 21:53
N. S.N. S.
105k7114210
$endgroup$
Hint Look for
$$K=beginbmatrix
cos(theta) & sin(theta) \
-sin(theta) & cos(theta)
endbmatrix$$
and write $beta$ in polar coordinates
$$beta=beginbmatrix
Rcos(phi) \
Rsin(phi)
endbmatrix$$
answered Mar 17 at 21:53
N. S.N. S.
105k7114210
answered Mar 17 at 21:53
N. S.N. S.
105k7114210
answered Mar 17 at 21:53
N. S.N. S.
105k7114210
answered Mar 17 at 21:53
N. S.N. S.
105k7114210
105k7114210
add a comment |
add a comment |
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2
$begingroup$
hints- try to do it for special values of $beta$, remind yourself what an orthogonal matrix represents geometrically
$endgroup$
– Calvin Khor
Mar 17 at 21:16
$begingroup$
I tried to show it for $beta =(c,0)'$ and $beta=(0,c)$ and I can find orthogonal matrices in these case. Can I ask you if its true or if I am looking for a counter example
$endgroup$
– user3503589
Mar 17 at 21:28
1
$begingroup$
So we can think of $K$ as a matrix which rotates $beta$ until it is on the $x$-axis?
$endgroup$
– user3503589
Mar 17 at 21:33
$begingroup$
Its true, and yes!
$endgroup$
– Calvin Khor
Mar 17 at 21:50