Changing $b$ of $AX =b$ [closed]
$begingroup$
Suppose 3 planes of the form $AX=b$ intersect at a unique point. Suppose in this case $b= (1,1,1)$. In case $b$ is changed to $(2,3,4)$ or any other vector, will the planes still intersect at a unique point?
linear-algebra systems-of-equations
edited Dec 7 '18 at 14:01
Andrews
3831317
asked Dec 7 '18 at 11:56
user46697user46697
196111
$endgroup$
closed as off-topic by GNUSupporter 8964民主女神 地下教會, Nosrati, Davide Giraudo, amWhy, KReiser Dec 8 '18 at 0:21
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Davide Giraudo, amWhy, KReiser
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Suppose 3 planes of the form $AX=b$ intersect at a unique point. Suppose in this case $b= (1,1,1)$. In case $b$ is changed to $(2,3,4)$ or any other vector, will the planes still intersect at a unique point?
linear-algebra systems-of-equations
edited Dec 7 '18 at 14:01
Andrews
3831317
asked Dec 7 '18 at 11:56
user46697user46697
196111
$endgroup$
closed as off-topic by GNUSupporter 8964民主女神 地下教會, Nosrati, Davide Giraudo, amWhy, KReiser Dec 8 '18 at 0:21
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Davide Giraudo, amWhy, KReiser
If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Is $A$ a vector or a matrix? As written, each row of $A$ corresponds to the coefficients of a plane. What are your thoughts?
$endgroup$
– Michael Burr
Dec 7 '18 at 12:01
add a comment |
$begingroup$
Suppose 3 planes of the form $AX=b$ intersect at a unique point. Suppose in this case $b= (1,1,1)$. In case $b$ is changed to $(2,3,4)$ or any other vector, will the planes still intersect at a unique point?
linear-algebra systems-of-equations
edited Dec 7 '18 at 14:01
Andrews
3831317
asked Dec 7 '18 at 11:56
user46697user46697
196111
$endgroup$
Suppose 3 planes of the form $AX=b$ intersect at a unique point. Suppose in this case $b= (1,1,1)$. In case $b$ is changed to $(2,3,4)$ or any other vector, will the planes still intersect at a unique point?
linear-algebra systems-of-equations
linear-algebra systems-of-equations
edited Dec 7 '18 at 14:01
Andrews
3831317
asked Dec 7 '18 at 11:56
user46697user46697
196111
edited Dec 7 '18 at 14:01
Andrews
3831317
asked Dec 7 '18 at 11:56
user46697user46697
196111
edited Dec 7 '18 at 14:01
Andrews
3831317
edited Dec 7 '18 at 14:01
Andrews
3831317
edited Dec 7 '18 at 14:01
Andrews
3831317
3831317
asked Dec 7 '18 at 11:56
user46697user46697
196111
asked Dec 7 '18 at 11:56
user46697user46697
196111
asked Dec 7 '18 at 11:56
user46697user46697
196111
196111
closed as off-topic by GNUSupporter 8964民主女神 地下教會, Nosrati, Davide Giraudo, amWhy, KReiser Dec 8 '18 at 0:21
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Davide Giraudo, amWhy, KReiser
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by GNUSupporter 8964民主女神 地下教會, Nosrati, Davide Giraudo, amWhy, KReiser Dec 8 '18 at 0:21
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Davide Giraudo, amWhy, KReiser
If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Is $A$ a vector or a matrix? As written, each row of $A$ corresponds to the coefficients of a plane. What are your thoughts?
$endgroup$
– Michael Burr
Dec 7 '18 at 12:01
add a comment |
2
$begingroup$
Is $A$ a vector or a matrix? As written, each row of $A$ corresponds to the coefficients of a plane. What are your thoughts?
$endgroup$
– Michael Burr
Dec 7 '18 at 12:01
2
2
$begingroup$
Is $A$ a vector or a matrix? As written, each row of $A$ corresponds to the coefficients of a plane. What are your thoughts?
$endgroup$
– Michael Burr
Dec 7 '18 at 12:01
$begingroup$
Is $A$ a vector or a matrix? As written, each row of $A$ corresponds to the coefficients of a plane. What are your thoughts?
$endgroup$
– Michael Burr
Dec 7 '18 at 12:01
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
In the first case the rank of $[A b]$ is $3$ (unique solution) and the system is consistent. So rank of $A $ is $3$. So any point/vector $binmathbb {R}^3$ can be expressed as a linear combination of columns of $A $. Equivalently there is a unique solution.
answered Dec 7 '18 at 12:08
AnyADAnyAD
2,088812
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
In the first case the rank of $[A b]$ is $3$ (unique solution) and the system is consistent. So rank of $A $ is $3$. So any point/vector $binmathbb {R}^3$ can be expressed as a linear combination of columns of $A $. Equivalently there is a unique solution.
answered Dec 7 '18 at 12:08
AnyADAnyAD
2,088812
$endgroup$
add a comment |
$begingroup$
In the first case the rank of $[A b]$ is $3$ (unique solution) and the system is consistent. So rank of $A $ is $3$. So any point/vector $binmathbb {R}^3$ can be expressed as a linear combination of columns of $A $. Equivalently there is a unique solution.
answered Dec 7 '18 at 12:08
AnyADAnyAD
2,088812
$endgroup$
add a comment |
$begingroup$
In the first case the rank of $[A b]$ is $3$ (unique solution) and the system is consistent. So rank of $A $ is $3$. So any point/vector $binmathbb {R}^3$ can be expressed as a linear combination of columns of $A $. Equivalently there is a unique solution.
answered Dec 7 '18 at 12:08
AnyADAnyAD
2,088812
$endgroup$
In the first case the rank of $[A b]$ is $3$ (unique solution) and the system is consistent. So rank of $A $ is $3$. So any point/vector $binmathbb {R}^3$ can be expressed as a linear combination of columns of $A $. Equivalently there is a unique solution.
answered Dec 7 '18 at 12:08
AnyADAnyAD
2,088812
answered Dec 7 '18 at 12:08
AnyADAnyAD
2,088812
answered Dec 7 '18 at 12:08
AnyADAnyAD
2,088812
answered Dec 7 '18 at 12:08
AnyADAnyAD
2,088812
2,088812
add a comment |
add a comment |
2
$begingroup$
Is $A$ a vector or a matrix? As written, each row of $A$ corresponds to the coefficients of a plane. What are your thoughts?
$endgroup$
– Michael Burr
Dec 7 '18 at 12:01