Linear algebra question about combination linear transformations
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Suppose that $T : M_{2×2}(Bbb R) rightarrow U_{2×2}(Bbb R)$ and $S : U_{2×2}(Bbb R) rightarrow P_3$ are linear transformations.
Then the composition map $S circ T : M_{2×2}(Bbb R) rightarrow P_3$ is never one-to-one.
To prove this is wrong, I have created a transformation $T$ and $S$ such that the standard matrix for the transformation $S circ T$ has a pivot in every column, and thus, one-to-one. However, I am not sure if I am on the right track or missing something, as this answer seems too easy.
linear-algebra matrices vector-spaces linear-transformations
edited Nov 18 at 23:47
Joey Kilpatrick
1,183422
asked Nov 18 at 23:42
question123
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up vote
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Suppose that $T : M_{2×2}(Bbb R) rightarrow U_{2×2}(Bbb R)$ and $S : U_{2×2}(Bbb R) rightarrow P_3$ are linear transformations.
Then the composition map $S circ T : M_{2×2}(Bbb R) rightarrow P_3$ is never one-to-one.
To prove this is wrong, I have created a transformation $T$ and $S$ such that the standard matrix for the transformation $S circ T$ has a pivot in every column, and thus, one-to-one. However, I am not sure if I am on the right track or missing something, as this answer seems too easy.
linear-algebra matrices vector-spaces linear-transformations
edited Nov 18 at 23:47
Joey Kilpatrick
1,183422
asked Nov 18 at 23:42
question123
11
Please help us to help you by explaining your notation. $M_{2times2}(Bbb{R})$ could well be the set of of $2 times 2$ matrices with elements in $Bbb{R}$ and $U_{2times2}(Bbb{R})$ might be the set of upper-triangular matrices. I can't hazard a guess about $P_3$. You need to tell us what the notations mean in your context.
– Rob Arthan
Nov 18 at 23:54
@RobArthan Hmm... Polynomials of degree at most 3? Then it would be $Bbb R^4to Bbb R^3toBbb R^4$.
– A.Γ.
Nov 19 at 0:07
@RobArthan U2×2(ℝ) is the set of upper triangular matrices and P3 is the set of polynomials of degree at most 3.
– question123
Nov 19 at 0:26
@A.Γ. Actually, I knew that with the obvious guess about the meaning $M$ and $U$ it didn't matter what $P_3$ meant, but the OP should have been more conscientious.
– Rob Arthan
Nov 19 at 1:03
@RobArthan I totally agree with you.
– A.Γ.
Nov 19 at 12:57
add a comment |
up vote
0
down vote
favorite
up vote
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down vote
favorite
Suppose that $T : M_{2×2}(Bbb R) rightarrow U_{2×2}(Bbb R)$ and $S : U_{2×2}(Bbb R) rightarrow P_3$ are linear transformations.
Then the composition map $S circ T : M_{2×2}(Bbb R) rightarrow P_3$ is never one-to-one.
To prove this is wrong, I have created a transformation $T$ and $S$ such that the standard matrix for the transformation $S circ T$ has a pivot in every column, and thus, one-to-one. However, I am not sure if I am on the right track or missing something, as this answer seems too easy.
linear-algebra matrices vector-spaces linear-transformations
edited Nov 18 at 23:47
Joey Kilpatrick
1,183422
asked Nov 18 at 23:42
question123
11
Suppose that $T : M_{2×2}(Bbb R) rightarrow U_{2×2}(Bbb R)$ and $S : U_{2×2}(Bbb R) rightarrow P_3$ are linear transformations.
Then the composition map $S circ T : M_{2×2}(Bbb R) rightarrow P_3$ is never one-to-one.
To prove this is wrong, I have created a transformation $T$ and $S$ such that the standard matrix for the transformation $S circ T$ has a pivot in every column, and thus, one-to-one. However, I am not sure if I am on the right track or missing something, as this answer seems too easy.
linear-algebra matrices vector-spaces linear-transformations
linear-algebra matrices vector-spaces linear-transformations
edited Nov 18 at 23:47
Joey Kilpatrick
1,183422
asked Nov 18 at 23:42
question123
11
edited Nov 18 at 23:47
Joey Kilpatrick
1,183422
asked Nov 18 at 23:42
question123
11
edited Nov 18 at 23:47
Joey Kilpatrick
1,183422
edited Nov 18 at 23:47
Joey Kilpatrick
1,183422
edited Nov 18 at 23:47
Joey Kilpatrick
1,183422
1,183422
asked Nov 18 at 23:42
question123
11
asked Nov 18 at 23:42
question123
11
asked Nov 18 at 23:42
question123
11
11
Please help us to help you by explaining your notation. $M_{2times2}(Bbb{R})$ could well be the set of of $2 times 2$ matrices with elements in $Bbb{R}$ and $U_{2times2}(Bbb{R})$ might be the set of upper-triangular matrices. I can't hazard a guess about $P_3$. You need to tell us what the notations mean in your context.
– Rob Arthan
Nov 18 at 23:54
@RobArthan Hmm... Polynomials of degree at most 3? Then it would be $Bbb R^4to Bbb R^3toBbb R^4$.
– A.Γ.
Nov 19 at 0:07
@RobArthan U2×2(ℝ) is the set of upper triangular matrices and P3 is the set of polynomials of degree at most 3.
– question123
Nov 19 at 0:26
@A.Γ. Actually, I knew that with the obvious guess about the meaning $M$ and $U$ it didn't matter what $P_3$ meant, but the OP should have been more conscientious.
– Rob Arthan
Nov 19 at 1:03
@RobArthan I totally agree with you.
– A.Γ.
Nov 19 at 12:57
add a comment |
Please help us to help you by explaining your notation. $M_{2times2}(Bbb{R})$ could well be the set of of $2 times 2$ matrices with elements in $Bbb{R}$ and $U_{2times2}(Bbb{R})$ might be the set of upper-triangular matrices. I can't hazard a guess about $P_3$. You need to tell us what the notations mean in your context.
– Rob Arthan
Nov 18 at 23:54
@RobArthan Hmm... Polynomials of degree at most 3? Then it would be $Bbb R^4to Bbb R^3toBbb R^4$.
– A.Γ.
Nov 19 at 0:07
@RobArthan U2×2(ℝ) is the set of upper triangular matrices and P3 is the set of polynomials of degree at most 3.
– question123
Nov 19 at 0:26
@A.Γ. Actually, I knew that with the obvious guess about the meaning $M$ and $U$ it didn't matter what $P_3$ meant, but the OP should have been more conscientious.
– Rob Arthan
Nov 19 at 1:03
@RobArthan I totally agree with you.
– A.Γ.
Nov 19 at 12:57
Please help us to help you by explaining your notation. $M_{2times2}(Bbb{R})$ could well be the set of of $2 times 2$ matrices with elements in $Bbb{R}$ and $U_{2times2}(Bbb{R})$ might be the set of upper-triangular matrices. I can't hazard a guess about $P_3$. You need to tell us what the notations mean in your context.
– Rob Arthan
Nov 18 at 23:54
Please help us to help you by explaining your notation. $M_{2times2}(Bbb{R})$ could well be the set of of $2 times 2$ matrices with elements in $Bbb{R}$ and $U_{2times2}(Bbb{R})$ might be the set of upper-triangular matrices. I can't hazard a guess about $P_3$. You need to tell us what the notations mean in your context.
– Rob Arthan
Nov 18 at 23:54
@RobArthan Hmm... Polynomials of degree at most 3? Then it would be $Bbb R^4to Bbb R^3toBbb R^4$.
– A.Γ.
Nov 19 at 0:07
@RobArthan Hmm... Polynomials of degree at most 3? Then it would be $Bbb R^4to Bbb R^3toBbb R^4$.
– A.Γ.
Nov 19 at 0:07
@RobArthan U2×2(ℝ) is the set of upper triangular matrices and P3 is the set of polynomials of degree at most 3.
– question123
Nov 19 at 0:26
@RobArthan U2×2(ℝ) is the set of upper triangular matrices and P3 is the set of polynomials of degree at most 3.
– question123
Nov 19 at 0:26
@A.Γ. Actually, I knew that with the obvious guess about the meaning $M$ and $U$ it didn't matter what $P_3$ meant, but the OP should have been more conscientious.
– Rob Arthan
Nov 19 at 1:03
@A.Γ. Actually, I knew that with the obvious guess about the meaning $M$ and $U$ it didn't matter what $P_3$ meant, but the OP should have been more conscientious.
– Rob Arthan
Nov 19 at 1:03
@RobArthan I totally agree with you.
– A.Γ.
Nov 19 at 12:57
@RobArthan I totally agree with you.
– A.Γ.
Nov 19 at 12:57
add a comment |
1 Answer
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0
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Following your comment clarifying the notation:
$M_{2times2}(Bbb{R})$ ($2times2$ matrices with real number entries) is a $4$-dimensional vector space over $Bbb{R}$. $U_{2times2}(Bbb{R})$ (upper-triangular $2times2$ matrices with real number entries) is a $3$-dimensional vector space over $Bbb{R}$.
Any linear transformation $T : M_{2times2}(Bbb{R}) to U_{2times2}(Bbb{R})$ must have a kernel of dimension at least $1$. So for any real vector space $V$ and any linear transformatiom $S : U_{2times2}(Bbb{R}) to V$, the composite $S circ T$ will have a non-trivial kernel and hence will not be one-one.
answered Nov 19 at 0:52
Rob Arthan
28.6k42865
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Following your comment clarifying the notation:
$M_{2times2}(Bbb{R})$ ($2times2$ matrices with real number entries) is a $4$-dimensional vector space over $Bbb{R}$. $U_{2times2}(Bbb{R})$ (upper-triangular $2times2$ matrices with real number entries) is a $3$-dimensional vector space over $Bbb{R}$.
Any linear transformation $T : M_{2times2}(Bbb{R}) to U_{2times2}(Bbb{R})$ must have a kernel of dimension at least $1$. So for any real vector space $V$ and any linear transformatiom $S : U_{2times2}(Bbb{R}) to V$, the composite $S circ T$ will have a non-trivial kernel and hence will not be one-one.
answered Nov 19 at 0:52
Rob Arthan
28.6k42865
add a comment |
up vote
0
down vote
Following your comment clarifying the notation:
$M_{2times2}(Bbb{R})$ ($2times2$ matrices with real number entries) is a $4$-dimensional vector space over $Bbb{R}$. $U_{2times2}(Bbb{R})$ (upper-triangular $2times2$ matrices with real number entries) is a $3$-dimensional vector space over $Bbb{R}$.
Any linear transformation $T : M_{2times2}(Bbb{R}) to U_{2times2}(Bbb{R})$ must have a kernel of dimension at least $1$. So for any real vector space $V$ and any linear transformatiom $S : U_{2times2}(Bbb{R}) to V$, the composite $S circ T$ will have a non-trivial kernel and hence will not be one-one.
answered Nov 19 at 0:52
Rob Arthan
28.6k42865
add a comment |
up vote
0
down vote
up vote
0
down vote
Following your comment clarifying the notation:
$M_{2times2}(Bbb{R})$ ($2times2$ matrices with real number entries) is a $4$-dimensional vector space over $Bbb{R}$. $U_{2times2}(Bbb{R})$ (upper-triangular $2times2$ matrices with real number entries) is a $3$-dimensional vector space over $Bbb{R}$.
Any linear transformation $T : M_{2times2}(Bbb{R}) to U_{2times2}(Bbb{R})$ must have a kernel of dimension at least $1$. So for any real vector space $V$ and any linear transformatiom $S : U_{2times2}(Bbb{R}) to V$, the composite $S circ T$ will have a non-trivial kernel and hence will not be one-one.
answered Nov 19 at 0:52
Rob Arthan
28.6k42865
Following your comment clarifying the notation:
$M_{2times2}(Bbb{R})$ ($2times2$ matrices with real number entries) is a $4$-dimensional vector space over $Bbb{R}$. $U_{2times2}(Bbb{R})$ (upper-triangular $2times2$ matrices with real number entries) is a $3$-dimensional vector space over $Bbb{R}$.
Any linear transformation $T : M_{2times2}(Bbb{R}) to U_{2times2}(Bbb{R})$ must have a kernel of dimension at least $1$. So for any real vector space $V$ and any linear transformatiom $S : U_{2times2}(Bbb{R}) to V$, the composite $S circ T$ will have a non-trivial kernel and hence will not be one-one.
answered Nov 19 at 0:52
Rob Arthan
28.6k42865
edited Nov 19 at 0:57
answered Nov 19 at 0:52
Rob Arthan
28.6k42865
answered Nov 19 at 0:52
Rob Arthan
28.6k42865
answered Nov 19 at 0:52
Rob Arthan
28.6k42865
28.6k42865
add a comment |
add a comment |
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Please help us to help you by explaining your notation. $M_{2times2}(Bbb{R})$ could well be the set of of $2 times 2$ matrices with elements in $Bbb{R}$ and $U_{2times2}(Bbb{R})$ might be the set of upper-triangular matrices. I can't hazard a guess about $P_3$. You need to tell us what the notations mean in your context.
– Rob Arthan
Nov 18 at 23:54
@RobArthan Hmm... Polynomials of degree at most 3? Then it would be $Bbb R^4to Bbb R^3toBbb R^4$.
– A.Γ.
Nov 19 at 0:07
@RobArthan U2×2(ℝ) is the set of upper triangular matrices and P3 is the set of polynomials of degree at most 3.
– question123
Nov 19 at 0:26
@A.Γ. Actually, I knew that with the obvious guess about the meaning $M$ and $U$ it didn't matter what $P_3$ meant, but the OP should have been more conscientious.
– Rob Arthan
Nov 19 at 1:03
@RobArthan I totally agree with you.
– A.Γ.
Nov 19 at 12:57