How to do the following change of Basis
$begingroup$
$V=mathbb{Q}^3$ The two following Bases of V are given
$S={(1,0,0) , (0,1,0) , (0,0,1)}, T={(-4,2,1) , (-1,0,1), (1,-1,1)}$;
Let $w ∈ V$ with $γ_T$$(w)$ =$ $$begin{pmatrix}1\0\1end{pmatrix}$. Determine $γ_S(w)$
Let $v ∈ V$ with $γ_S$$(v)$ =$ $$begin{pmatrix}3\-2\1end{pmatrix}$. Determine $γ_T(v)$
"How would I begin to solve this problem/what steps should be taken to solve it."
linear-algebra change-of-basis
edited Dec 14 '18 at 3:12
Key Flex
8,00261233
asked Dec 14 '18 at 3:05
DanDan
61
$endgroup$
add a comment |
$begingroup$
$V=mathbb{Q}^3$ The two following Bases of V are given
$S={(1,0,0) , (0,1,0) , (0,0,1)}, T={(-4,2,1) , (-1,0,1), (1,-1,1)}$;
Let $w ∈ V$ with $γ_T$$(w)$ =$ $$begin{pmatrix}1\0\1end{pmatrix}$. Determine $γ_S(w)$
Let $v ∈ V$ with $γ_S$$(v)$ =$ $$begin{pmatrix}3\-2\1end{pmatrix}$. Determine $γ_T(v)$
"How would I begin to solve this problem/what steps should be taken to solve it."
linear-algebra change-of-basis
edited Dec 14 '18 at 3:12
Key Flex
8,00261233
asked Dec 14 '18 at 3:05
DanDan
61
$endgroup$
$begingroup$
Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
$endgroup$
– T. Fo
Dec 14 '18 at 3:40
add a comment |
$begingroup$
$V=mathbb{Q}^3$ The two following Bases of V are given
$S={(1,0,0) , (0,1,0) , (0,0,1)}, T={(-4,2,1) , (-1,0,1), (1,-1,1)}$;
Let $w ∈ V$ with $γ_T$$(w)$ =$ $$begin{pmatrix}1\0\1end{pmatrix}$. Determine $γ_S(w)$
Let $v ∈ V$ with $γ_S$$(v)$ =$ $$begin{pmatrix}3\-2\1end{pmatrix}$. Determine $γ_T(v)$
"How would I begin to solve this problem/what steps should be taken to solve it."
linear-algebra change-of-basis
edited Dec 14 '18 at 3:12
Key Flex
8,00261233
asked Dec 14 '18 at 3:05
DanDan
61
$endgroup$
$V=mathbb{Q}^3$ The two following Bases of V are given
$S={(1,0,0) , (0,1,0) , (0,0,1)}, T={(-4,2,1) , (-1,0,1), (1,-1,1)}$;
Let $w ∈ V$ with $γ_T$$(w)$ =$ $$begin{pmatrix}1\0\1end{pmatrix}$. Determine $γ_S(w)$
Let $v ∈ V$ with $γ_S$$(v)$ =$ $$begin{pmatrix}3\-2\1end{pmatrix}$. Determine $γ_T(v)$
"How would I begin to solve this problem/what steps should be taken to solve it."
linear-algebra change-of-basis
linear-algebra change-of-basis
edited Dec 14 '18 at 3:12
Key Flex
8,00261233
asked Dec 14 '18 at 3:05
DanDan
61
edited Dec 14 '18 at 3:12
Key Flex
8,00261233
asked Dec 14 '18 at 3:05
DanDan
61
edited Dec 14 '18 at 3:12
Key Flex
8,00261233
edited Dec 14 '18 at 3:12
Key Flex
8,00261233
edited Dec 14 '18 at 3:12
Key Flex
8,00261233
8,00261233
asked Dec 14 '18 at 3:05
DanDan
61
asked Dec 14 '18 at 3:05
DanDan
61
asked Dec 14 '18 at 3:05
DanDan
61
61
$begingroup$
Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
$endgroup$
– T. Fo
Dec 14 '18 at 3:40
add a comment |
$begingroup$
Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
$endgroup$
– T. Fo
Dec 14 '18 at 3:40
$begingroup$
Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
$endgroup$
– T. Fo
Dec 14 '18 at 3:40
$begingroup$
Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
$endgroup$
– T. Fo
Dec 14 '18 at 3:40
add a comment |
2 Answers
2
active
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$begingroup$
Hint: You need to find the change of basis matrix. For the $T rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.
answered Dec 14 '18 at 3:22
T. FoT. Fo
466311
$endgroup$
add a comment |
$begingroup$
$gamma_T rightarrow gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $gamma_S(w)$ you simply add the components of these vectors.
$gamma_T rightarrow gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $gamma_S$ components to $gamma_T$ components.
answered Dec 14 '18 at 15:18
gandalf61gandalf61
8,649725
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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
$begingroup$
Hint: You need to find the change of basis matrix. For the $T rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.
answered Dec 14 '18 at 3:22
T. FoT. Fo
466311
$endgroup$
add a comment |
$begingroup$
Hint: You need to find the change of basis matrix. For the $T rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.
answered Dec 14 '18 at 3:22
T. FoT. Fo
466311
$endgroup$
add a comment |
$begingroup$
Hint: You need to find the change of basis matrix. For the $T rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.
answered Dec 14 '18 at 3:22
T. FoT. Fo
466311
$endgroup$
Hint: You need to find the change of basis matrix. For the $T rightarrow S$ direction, find $[ [t_1]_s, [t_2]_s, [t_3]_s ]$ and then right multiply this by the coordinate vector already given to you.
answered Dec 14 '18 at 3:22
T. FoT. Fo
466311
answered Dec 14 '18 at 3:22
T. FoT. Fo
466311
answered Dec 14 '18 at 3:22
T. FoT. Fo
466311
answered Dec 14 '18 at 3:22
T. FoT. Fo
466311
466311
add a comment |
add a comment |
$begingroup$
$gamma_T rightarrow gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $gamma_S(w)$ you simply add the components of these vectors.
$gamma_T rightarrow gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $gamma_S$ components to $gamma_T$ components.
answered Dec 14 '18 at 15:18
gandalf61gandalf61
8,649725
$endgroup$
add a comment |
$begingroup$
$gamma_T rightarrow gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $gamma_S(w)$ you simply add the components of these vectors.
$gamma_T rightarrow gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $gamma_S$ components to $gamma_T$ components.
answered Dec 14 '18 at 15:18
gandalf61gandalf61
8,649725
$endgroup$
add a comment |
$begingroup$
$gamma_T rightarrow gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $gamma_S(w)$ you simply add the components of these vectors.
$gamma_T rightarrow gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $gamma_S$ components to $gamma_T$ components.
answered Dec 14 '18 at 15:18
gandalf61gandalf61
8,649725
$endgroup$
$gamma_T rightarrow gamma_S$ is straightforward, since you are given the components of the basis vectors of $T$ with respect to $S$. $w$ is the sum of the first and third basis vectors of $T$ so to find $gamma_S(w)$ you simply add the components of these vectors.
$gamma_T rightarrow gamma_S$ is a little more difficult. You have to find the components of the basis vectors of $S$ with respect to $T$. This is equivalent to inverting the matrix that transforms $gamma_S$ components to $gamma_T$ components.
answered Dec 14 '18 at 15:18
gandalf61gandalf61
8,649725
answered Dec 14 '18 at 15:18
gandalf61gandalf61
8,649725
answered Dec 14 '18 at 15:18
gandalf61gandalf61
8,649725
answered Dec 14 '18 at 15:18
gandalf61gandalf61
8,649725
8,649725
add a comment |
add a comment |
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$begingroup$
Someone else just asked a question explaining how to find the change of basis matrix, if you don't know how to do that check it out here: math.stackexchange.com/q/3038872/525520
$endgroup$
– T. Fo
Dec 14 '18 at 3:40