Mixing
How to Define the Display Matrix from two linear mappings
$begingroup$
I'm working right now on this task, but i dont't know how to start,
Im glad for every hint, I hope we can solve this task together
$$ Given , are , the , following , ordered , Bases , of , mathbb{R²}, mathbb{R³} , and , mathbb{R⁴}$$
$$
B₂=
begin{pmatrix}
1 \
0
end{pmatrix}
,
begin{pmatrix}
0 \
1
end{pmatrix}
quad
B₃=
begin{pmatrix}
1 \
1 \
1 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
0 \
end{pmatrix}
,
begin{pmatrix}
1 \
0 \
0 \
end{pmatrix}
quad
B₄=
begin{pmatrix}
1 \
1 \
1 \
1 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
1 \
0 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
0 \
0 \end{pmatrix}
,
begin{pmatrix}
1 \
0 \
0 \
0 \end{pmatrix}
$$
$$ Let , us , consider , now , two , linear , mappings , varphi:mathbb{R²} → mathbb{R³} , , psi: mathbb{R³} , → , mathbb{R⁴} $$
$$
varphi
begin{pmatrix}
v1 \
v2
end{pmatrix}
=
begin{pmatrix}
v1 - v2 \
0 \
2v1 - v2
end{pmatrix}
quad
and , psi
begin{pmatrix}
v1\
v2 \
v3 \
end{pmatrix}
=
begin{pmatrix}
v₁ + 2v₃ \
v₂ - v₃ \
v₁ + v₂ \
2v₁ + 3v₃ \
end{pmatrix}
$$
$$ Define , the , display , matrix , M
begin{matrix}
B₂ \
B₃
end{matrix}
(varphi), , M
begin{matrix}
B₃ \
B₄
end{matrix}
(psi) , and , M
begin{matrix}
B₂ \
B₄
end{matrix}
(psi , ○ , varphi)
$$
Thank you for any hint :)
Definition of the display matrix
linear-algebra matrices mathematica popular-math
asked Jan 26 at 16:26
deandean
11
$endgroup$
add a comment |
$begingroup$
I'm working right now on this task, but i dont't know how to start,
Im glad for every hint, I hope we can solve this task together
$$ Given , are , the , following , ordered , Bases , of , mathbb{R²}, mathbb{R³} , and , mathbb{R⁴}$$
$$
B₂=
begin{pmatrix}
1 \
0
end{pmatrix}
,
begin{pmatrix}
0 \
1
end{pmatrix}
quad
B₃=
begin{pmatrix}
1 \
1 \
1 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
0 \
end{pmatrix}
,
begin{pmatrix}
1 \
0 \
0 \
end{pmatrix}
quad
B₄=
begin{pmatrix}
1 \
1 \
1 \
1 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
1 \
0 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
0 \
0 \end{pmatrix}
,
begin{pmatrix}
1 \
0 \
0 \
0 \end{pmatrix}
$$
$$ Let , us , consider , now , two , linear , mappings , varphi:mathbb{R²} → mathbb{R³} , , psi: mathbb{R³} , → , mathbb{R⁴} $$
$$
varphi
begin{pmatrix}
v1 \
v2
end{pmatrix}
=
begin{pmatrix}
v1 - v2 \
0 \
2v1 - v2
end{pmatrix}
quad
and , psi
begin{pmatrix}
v1\
v2 \
v3 \
end{pmatrix}
=
begin{pmatrix}
v₁ + 2v₃ \
v₂ - v₃ \
v₁ + v₂ \
2v₁ + 3v₃ \
end{pmatrix}
$$
$$ Define , the , display , matrix , M
begin{matrix}
B₂ \
B₃
end{matrix}
(varphi), , M
begin{matrix}
B₃ \
B₄
end{matrix}
(psi) , and , M
begin{matrix}
B₂ \
B₄
end{matrix}
(psi , ○ , varphi)
$$
Thank you for any hint :)
Definition of the display matrix
linear-algebra matrices mathematica popular-math
asked Jan 26 at 16:26
deandean
11
$endgroup$
$begingroup$
Do you know the definition of a display matrix?
$endgroup$
– user635162
Jan 26 at 17:24
$begingroup$
Yes sure, I added it as a picture in the main question.
$endgroup$
– dean
Jan 26 at 18:02
$begingroup$
Then just apply it. Put each basis vector into the function and write the corresponding image as a linear combination of the other basis. The coefficients of that linear combination are then the entries of the matrix, just as the definition states.
$endgroup$
– user635162
Jan 26 at 18:08
add a comment |
$begingroup$
I'm working right now on this task, but i dont't know how to start,
Im glad for every hint, I hope we can solve this task together
$$ Given , are , the , following , ordered , Bases , of , mathbb{R²}, mathbb{R³} , and , mathbb{R⁴}$$
$$
B₂=
begin{pmatrix}
1 \
0
end{pmatrix}
,
begin{pmatrix}
0 \
1
end{pmatrix}
quad
B₃=
begin{pmatrix}
1 \
1 \
1 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
0 \
end{pmatrix}
,
begin{pmatrix}
1 \
0 \
0 \
end{pmatrix}
quad
B₄=
begin{pmatrix}
1 \
1 \
1 \
1 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
1 \
0 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
0 \
0 \end{pmatrix}
,
begin{pmatrix}
1 \
0 \
0 \
0 \end{pmatrix}
$$
$$ Let , us , consider , now , two , linear , mappings , varphi:mathbb{R²} → mathbb{R³} , , psi: mathbb{R³} , → , mathbb{R⁴} $$
$$
varphi
begin{pmatrix}
v1 \
v2
end{pmatrix}
=
begin{pmatrix}
v1 - v2 \
0 \
2v1 - v2
end{pmatrix}
quad
and , psi
begin{pmatrix}
v1\
v2 \
v3 \
end{pmatrix}
=
begin{pmatrix}
v₁ + 2v₃ \
v₂ - v₃ \
v₁ + v₂ \
2v₁ + 3v₃ \
end{pmatrix}
$$
$$ Define , the , display , matrix , M
begin{matrix}
B₂ \
B₃
end{matrix}
(varphi), , M
begin{matrix}
B₃ \
B₄
end{matrix}
(psi) , and , M
begin{matrix}
B₂ \
B₄
end{matrix}
(psi , ○ , varphi)
$$
Thank you for any hint :)
Definition of the display matrix
linear-algebra matrices mathematica popular-math
asked Jan 26 at 16:26
deandean
11
$endgroup$
I'm working right now on this task, but i dont't know how to start,
Im glad for every hint, I hope we can solve this task together
$$ Given , are , the , following , ordered , Bases , of , mathbb{R²}, mathbb{R³} , and , mathbb{R⁴}$$
$$
B₂=
begin{pmatrix}
1 \
0
end{pmatrix}
,
begin{pmatrix}
0 \
1
end{pmatrix}
quad
B₃=
begin{pmatrix}
1 \
1 \
1 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
0 \
end{pmatrix}
,
begin{pmatrix}
1 \
0 \
0 \
end{pmatrix}
quad
B₄=
begin{pmatrix}
1 \
1 \
1 \
1 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
1 \
0 \
end{pmatrix}
,
begin{pmatrix}
1 \
1 \
0 \
0 \end{pmatrix}
,
begin{pmatrix}
1 \
0 \
0 \
0 \end{pmatrix}
$$
$$ Let , us , consider , now , two , linear , mappings , varphi:mathbb{R²} → mathbb{R³} , , psi: mathbb{R³} , → , mathbb{R⁴} $$
$$
varphi
begin{pmatrix}
v1 \
v2
end{pmatrix}
=
begin{pmatrix}
v1 - v2 \
0 \
2v1 - v2
end{pmatrix}
quad
and , psi
begin{pmatrix}
v1\
v2 \
v3 \
end{pmatrix}
=
begin{pmatrix}
v₁ + 2v₃ \
v₂ - v₃ \
v₁ + v₂ \
2v₁ + 3v₃ \
end{pmatrix}
$$
$$ Define , the , display , matrix , M
begin{matrix}
B₂ \
B₃
end{matrix}
(varphi), , M
begin{matrix}
B₃ \
B₄
end{matrix}
(psi) , and , M
begin{matrix}
B₂ \
B₄
end{matrix}
(psi , ○ , varphi)
$$
Thank you for any hint :)
Definition of the display matrix
linear-algebra matrices mathematica popular-math
linear-algebra matrices mathematica popular-math
asked Jan 26 at 16:26
deandean
11
asked Jan 26 at 16:26
deandean
11
edited Jan 26 at 18:02
dean
asked Jan 26 at 16:26
deandean
11
asked Jan 26 at 16:26
deandean
11
asked Jan 26 at 16:26
deandean
11
11
$begingroup$
Do you know the definition of a display matrix?
$endgroup$
– user635162
Jan 26 at 17:24
$begingroup$
Yes sure, I added it as a picture in the main question.
$endgroup$
– dean
Jan 26 at 18:02
$begingroup$
Then just apply it. Put each basis vector into the function and write the corresponding image as a linear combination of the other basis. The coefficients of that linear combination are then the entries of the matrix, just as the definition states.
$endgroup$
– user635162
Jan 26 at 18:08
add a comment |
$begingroup$
Do you know the definition of a display matrix?
$endgroup$
– user635162
Jan 26 at 17:24
$begingroup$
Yes sure, I added it as a picture in the main question.
$endgroup$
– dean
Jan 26 at 18:02
$begingroup$
Then just apply it. Put each basis vector into the function and write the corresponding image as a linear combination of the other basis. The coefficients of that linear combination are then the entries of the matrix, just as the definition states.
$endgroup$
– user635162
Jan 26 at 18:08
$begingroup$
Do you know the definition of a display matrix?
$endgroup$
– user635162
Jan 26 at 17:24
$begingroup$
Do you know the definition of a display matrix?
$endgroup$
– user635162
Jan 26 at 17:24
$begingroup$
Yes sure, I added it as a picture in the main question.
$endgroup$
– dean
Jan 26 at 18:02
$begingroup$
Yes sure, I added it as a picture in the main question.
$endgroup$
– dean
Jan 26 at 18:02
$begingroup$
Then just apply it. Put each basis vector into the function and write the corresponding image as a linear combination of the other basis. The coefficients of that linear combination are then the entries of the matrix, just as the definition states.
$endgroup$
– user635162
Jan 26 at 18:08
$begingroup$
Then just apply it. Put each basis vector into the function and write the corresponding image as a linear combination of the other basis. The coefficients of that linear combination are then the entries of the matrix, just as the definition states.
$endgroup$
– user635162
Jan 26 at 18:08
add a comment |
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$begingroup$
Do you know the definition of a display matrix?
$endgroup$
– user635162
Jan 26 at 17:24
$begingroup$
Yes sure, I added it as a picture in the main question.
$endgroup$
– dean
Jan 26 at 18:02
$begingroup$
Then just apply it. Put each basis vector into the function and write the corresponding image as a linear combination of the other basis. The coefficients of that linear combination are then the entries of the matrix, just as the definition states.
$endgroup$
– user635162
Jan 26 at 18:08