Mixing
If b is a linear combination then Ax=b has a solution
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Q. True or False? If A = [a1, a2, a3], where aj is the jth column of A, and if b is a linear combination of a1 and a3 then Ax=b has a solution.
I think its false because if b was a linear combination of only a1 and a3 wouldn't that only be finding the solution to the matrix A = [a1, a3]?
linear-algebra matrices
asked Jan 26 at 16:46
PeraltaLearnsPeraltaLearns
205
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add a comment |
$begingroup$
Q. True or False? If A = [a1, a2, a3], where aj is the jth column of A, and if b is a linear combination of a1 and a3 then Ax=b has a solution.
I think its false because if b was a linear combination of only a1 and a3 wouldn't that only be finding the solution to the matrix A = [a1, a3]?
linear-algebra matrices
asked Jan 26 at 16:46
PeraltaLearnsPeraltaLearns
205
$endgroup$
$begingroup$
No it's true; the criterion for such a linear system to have solutions is that the augmented matrix has the same rank as $A$.
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– Bernard
Jan 26 at 17:02
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Look at the definitions! If $b$ is a linear combination of $a_1$ and $a_3$ then it follows that $b$ is a linear combination of $a_1$, $a_2$ and $a_3$.
$endgroup$
– David C. Ullrich
Jan 26 at 17:39
add a comment |
$begingroup$
Q. True or False? If A = [a1, a2, a3], where aj is the jth column of A, and if b is a linear combination of a1 and a3 then Ax=b has a solution.
I think its false because if b was a linear combination of only a1 and a3 wouldn't that only be finding the solution to the matrix A = [a1, a3]?
linear-algebra matrices
asked Jan 26 at 16:46
PeraltaLearnsPeraltaLearns
205
$endgroup$
Q. True or False? If A = [a1, a2, a3], where aj is the jth column of A, and if b is a linear combination of a1 and a3 then Ax=b has a solution.
I think its false because if b was a linear combination of only a1 and a3 wouldn't that only be finding the solution to the matrix A = [a1, a3]?
linear-algebra matrices
linear-algebra matrices
asked Jan 26 at 16:46
PeraltaLearnsPeraltaLearns
205
asked Jan 26 at 16:46
PeraltaLearnsPeraltaLearns
205
asked Jan 26 at 16:46
PeraltaLearnsPeraltaLearns
205
asked Jan 26 at 16:46
PeraltaLearnsPeraltaLearns
205
asked Jan 26 at 16:46
PeraltaLearnsPeraltaLearns
205
205
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No it's true; the criterion for such a linear system to have solutions is that the augmented matrix has the same rank as $A$.
$endgroup$
– Bernard
Jan 26 at 17:02
$begingroup$
Look at the definitions! If $b$ is a linear combination of $a_1$ and $a_3$ then it follows that $b$ is a linear combination of $a_1$, $a_2$ and $a_3$.
$endgroup$
– David C. Ullrich
Jan 26 at 17:39
add a comment |
$begingroup$
No it's true; the criterion for such a linear system to have solutions is that the augmented matrix has the same rank as $A$.
$endgroup$
– Bernard
Jan 26 at 17:02
$begingroup$
Look at the definitions! If $b$ is a linear combination of $a_1$ and $a_3$ then it follows that $b$ is a linear combination of $a_1$, $a_2$ and $a_3$.
$endgroup$
– David C. Ullrich
Jan 26 at 17:39
$begingroup$
No it's true; the criterion for such a linear system to have solutions is that the augmented matrix has the same rank as $A$.
$endgroup$
– Bernard
Jan 26 at 17:02
$begingroup$
No it's true; the criterion for such a linear system to have solutions is that the augmented matrix has the same rank as $A$.
$endgroup$
– Bernard
Jan 26 at 17:02
$begingroup$
Look at the definitions! If $b$ is a linear combination of $a_1$ and $a_3$ then it follows that $b$ is a linear combination of $a_1$, $a_2$ and $a_3$.
$endgroup$
– David C. Ullrich
Jan 26 at 17:39
$begingroup$
Look at the definitions! If $b$ is a linear combination of $a_1$ and $a_3$ then it follows that $b$ is a linear combination of $a_1$, $a_2$ and $a_3$.
$endgroup$
– David C. Ullrich
Jan 26 at 17:39
add a comment |
1 Answer
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$begingroup$
Note that
$$A x = A begin{bmatrix}x_1\x_2\x_3end{bmatrix} = x_1a_1 + x_2a_2 + x_3a_3 $$
If $b$ is a linear combination of $a_1$ and $a_3$, then it is also a linear combination of $a_1$, $a_2$ and $a_3$, since
$$b = lambda_1 a_1 + lambda_2 a_3 = lambda_1 a_1 + lambda_2 a_3 + 0 cdot a_2 $$
This means that
$$x := begin{bmatrix}lambda_1\0\lambda_2end{bmatrix}$$
is a solution to
$$Ax = b $$
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
oldest
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votes
$begingroup$
Note that
$$A x = A begin{bmatrix}x_1\x_2\x_3end{bmatrix} = x_1a_1 + x_2a_2 + x_3a_3 $$
If $b$ is a linear combination of $a_1$ and $a_3$, then it is also a linear combination of $a_1$, $a_2$ and $a_3$, since
$$b = lambda_1 a_1 + lambda_2 a_3 = lambda_1 a_1 + lambda_2 a_3 + 0 cdot a_2 $$
This means that
$$x := begin{bmatrix}lambda_1\0\lambda_2end{bmatrix}$$
is a solution to
$$Ax = b $$
$endgroup$
add a comment |
$begingroup$
Note that
$$A x = A begin{bmatrix}x_1\x_2\x_3end{bmatrix} = x_1a_1 + x_2a_2 + x_3a_3 $$
If $b$ is a linear combination of $a_1$ and $a_3$, then it is also a linear combination of $a_1$, $a_2$ and $a_3$, since
$$b = lambda_1 a_1 + lambda_2 a_3 = lambda_1 a_1 + lambda_2 a_3 + 0 cdot a_2 $$
This means that
$$x := begin{bmatrix}lambda_1\0\lambda_2end{bmatrix}$$
is a solution to
$$Ax = b $$
$endgroup$
add a comment |
$begingroup$
Note that
$$A x = A begin{bmatrix}x_1\x_2\x_3end{bmatrix} = x_1a_1 + x_2a_2 + x_3a_3 $$
If $b$ is a linear combination of $a_1$ and $a_3$, then it is also a linear combination of $a_1$, $a_2$ and $a_3$, since
$$b = lambda_1 a_1 + lambda_2 a_3 = lambda_1 a_1 + lambda_2 a_3 + 0 cdot a_2 $$
This means that
$$x := begin{bmatrix}lambda_1\0\lambda_2end{bmatrix}$$
is a solution to
$$Ax = b $$
$endgroup$
Note that
$$A x = A begin{bmatrix}x_1\x_2\x_3end{bmatrix} = x_1a_1 + x_2a_2 + x_3a_3 $$
If $b$ is a linear combination of $a_1$ and $a_3$, then it is also a linear combination of $a_1$, $a_2$ and $a_3$, since
$$b = lambda_1 a_1 + lambda_2 a_3 = lambda_1 a_1 + lambda_2 a_3 + 0 cdot a_2 $$
This means that
$$x := begin{bmatrix}lambda_1\0\lambda_2end{bmatrix}$$
is a solution to
$$Ax = b $$
edited Jan 26 at 17:06
answered Jan 26 at 16:50
user635162
add a comment |
add a comment |
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$begingroup$
No it's true; the criterion for such a linear system to have solutions is that the augmented matrix has the same rank as $A$.
$endgroup$
– Bernard
Jan 26 at 17:02
$begingroup$
Look at the definitions! If $b$ is a linear combination of $a_1$ and $a_3$ then it follows that $b$ is a linear combination of $a_1$, $a_2$ and $a_3$.
$endgroup$
– David C. Ullrich
Jan 26 at 17:39