Finding Coordinate Vector?
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In a new basis of $mathbb R^2$ , the coordinate vector of the vector $[2,3]$ is $[4,3]$ and that of the vector $[4,5]$ is $[6,6]$. Given this information, what is the coordinate vector of $[6,7]$?
vectors
edited Dec 7 '18 at 9:12
José Carlos Santos
159k22126231
asked Dec 7 '18 at 9:05
user9605051user9605051
1
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add a comment |
$begingroup$
In a new basis of $mathbb R^2$ , the coordinate vector of the vector $[2,3]$ is $[4,3]$ and that of the vector $[4,5]$ is $[6,6]$. Given this information, what is the coordinate vector of $[6,7]$?
vectors
edited Dec 7 '18 at 9:12
José Carlos Santos
159k22126231
asked Dec 7 '18 at 9:05
user9605051user9605051
1
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 7 '18 at 9:13
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$begingroup$
In a new basis of $mathbb R^2$ , the coordinate vector of the vector $[2,3]$ is $[4,3]$ and that of the vector $[4,5]$ is $[6,6]$. Given this information, what is the coordinate vector of $[6,7]$?
vectors
edited Dec 7 '18 at 9:12
José Carlos Santos
159k22126231
asked Dec 7 '18 at 9:05
user9605051user9605051
1
$endgroup$
In a new basis of $mathbb R^2$ , the coordinate vector of the vector $[2,3]$ is $[4,3]$ and that of the vector $[4,5]$ is $[6,6]$. Given this information, what is the coordinate vector of $[6,7]$?
vectors
vectors
edited Dec 7 '18 at 9:12
José Carlos Santos
159k22126231
asked Dec 7 '18 at 9:05
user9605051user9605051
1
edited Dec 7 '18 at 9:12
José Carlos Santos
159k22126231
asked Dec 7 '18 at 9:05
user9605051user9605051
1
edited Dec 7 '18 at 9:12
José Carlos Santos
159k22126231
edited Dec 7 '18 at 9:12
José Carlos Santos
159k22126231
edited Dec 7 '18 at 9:12
José Carlos Santos
159k22126231
159k22126231
asked Dec 7 '18 at 9:05
user9605051user9605051
1
asked Dec 7 '18 at 9:05
user9605051user9605051
1
asked Dec 7 '18 at 9:05
user9605051user9605051
1
1
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 7 '18 at 9:13
add a comment |
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 7 '18 at 9:13
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 7 '18 at 9:13
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 7 '18 at 9:13
add a comment |
2 Answers
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It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.
answered Dec 7 '18 at 9:07
Kavi Rama MurthyKavi Rama Murthy
58.2k42161
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The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.
If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $mathbf{e_1}$ and $mathbf{e_2}$. Then
$4mathbf{e_1}+3mathbf{e_2}=[2,3]$
$6mathbf{e_1}+6mathbf{e_2}=[4,5]$
$Rightarrow 2mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$
$Rightarrow mathbf{e_1} = left[0, frac 12 right], space mathbf{e_2} = left[frac 23, frac 13 right]$
$ Rightarrow [6,7] = 8mathbf{e_1} + 9mathbf{e_2}$
answered Dec 7 '18 at 9:30
gandalf61gandalf61
8,554725
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
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$begingroup$
It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.
answered Dec 7 '18 at 9:07
Kavi Rama MurthyKavi Rama Murthy
58.2k42161
$endgroup$
add a comment |
$begingroup$
It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.
answered Dec 7 '18 at 9:07
Kavi Rama MurthyKavi Rama Murthy
58.2k42161
$endgroup$
add a comment |
$begingroup$
It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.
answered Dec 7 '18 at 9:07
Kavi Rama MurthyKavi Rama Murthy
58.2k42161
$endgroup$
It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.
answered Dec 7 '18 at 9:07
Kavi Rama MurthyKavi Rama Murthy
58.2k42161
answered Dec 7 '18 at 9:07
Kavi Rama MurthyKavi Rama Murthy
58.2k42161
answered Dec 7 '18 at 9:07
Kavi Rama MurthyKavi Rama Murthy
58.2k42161
answered Dec 7 '18 at 9:07
Kavi Rama MurthyKavi Rama Murthy
58.2k42161
58.2k42161
add a comment |
add a comment |
$begingroup$
The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.
If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $mathbf{e_1}$ and $mathbf{e_2}$. Then
$4mathbf{e_1}+3mathbf{e_2}=[2,3]$
$6mathbf{e_1}+6mathbf{e_2}=[4,5]$
$Rightarrow 2mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$
$Rightarrow mathbf{e_1} = left[0, frac 12 right], space mathbf{e_2} = left[frac 23, frac 13 right]$
$ Rightarrow [6,7] = 8mathbf{e_1} + 9mathbf{e_2}$
answered Dec 7 '18 at 9:30
gandalf61gandalf61
8,554725
$endgroup$
add a comment |
$begingroup$
The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.
If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $mathbf{e_1}$ and $mathbf{e_2}$. Then
$4mathbf{e_1}+3mathbf{e_2}=[2,3]$
$6mathbf{e_1}+6mathbf{e_2}=[4,5]$
$Rightarrow 2mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$
$Rightarrow mathbf{e_1} = left[0, frac 12 right], space mathbf{e_2} = left[frac 23, frac 13 right]$
$ Rightarrow [6,7] = 8mathbf{e_1} + 9mathbf{e_2}$
answered Dec 7 '18 at 9:30
gandalf61gandalf61
8,554725
$endgroup$
add a comment |
$begingroup$
The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.
If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $mathbf{e_1}$ and $mathbf{e_2}$. Then
$4mathbf{e_1}+3mathbf{e_2}=[2,3]$
$6mathbf{e_1}+6mathbf{e_2}=[4,5]$
$Rightarrow 2mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$
$Rightarrow mathbf{e_1} = left[0, frac 12 right], space mathbf{e_2} = left[frac 23, frac 13 right]$
$ Rightarrow [6,7] = 8mathbf{e_1} + 9mathbf{e_2}$
answered Dec 7 '18 at 9:30
gandalf61gandalf61
8,554725
$endgroup$
The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.
If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $mathbf{e_1}$ and $mathbf{e_2}$. Then
$4mathbf{e_1}+3mathbf{e_2}=[2,3]$
$6mathbf{e_1}+6mathbf{e_2}=[4,5]$
$Rightarrow 2mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$
$Rightarrow mathbf{e_1} = left[0, frac 12 right], space mathbf{e_2} = left[frac 23, frac 13 right]$
$ Rightarrow [6,7] = 8mathbf{e_1} + 9mathbf{e_2}$
answered Dec 7 '18 at 9:30
gandalf61gandalf61
8,554725
answered Dec 7 '18 at 9:30
gandalf61gandalf61
8,554725
answered Dec 7 '18 at 9:30
gandalf61gandalf61
8,554725
answered Dec 7 '18 at 9:30
gandalf61gandalf61
8,554725
8,554725
add a comment |
add a comment |
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$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Dec 7 '18 at 9:13