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For which scalars a, b, c, d do the solutions to the equation ax+by+cz=d form a subspace of R(3).

Mathematics
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In \(ax + by + cz = d\), if x and y are any number in R(1), then z is constrained to \[z=\frac{d-ax-by}{c}\] The subspace S = { any (x, y, z)} is a plane which does not pass through (0,0,0) and is therefore not a subspace.
Actually, I failed to address all the trivial cases. If a = b = c = 0, there is no solution. The empty set is not a subspace. If one or two of a,b, or c equals 0, then you can rotate your axes such that the one that is not 0 is the constant for z, in which case the theorem holds.
so your saying that if atleast a, b, or c is 0 then the others can be non zero numbers and be a subspace of R(3)?

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No, I'm saying that if d = 0 then the solution set (x, y, z) is a subspace of R(3) UNLESS all three are equal to 0.
ll three a, b, c****
and that would be the same thing if it was a, b, c to the equation ax+by=c form a subspace of R(2)... then c=0 in this case.
also, if d=0, then as long as atleast one of a, b, or c is non zero then it IS a subspace right?
actually the set u r given is a subspace if and only if d=0.
but why? and does it matter what a, b, and c are?
no.
u have \[ \large D=\{(x,y,z):ax+by+cz=d\} \]
if d=0 then u have \[ \large D=\{(x,y,z):ax+by+cz=0\}. \] So let \(u=(x_0,y_0,z_0)\), \(v=(x_1,y_1,z_1)\) both in \(D\) and \(\alpha\in\mathbb{R}\) so (i) \(u+v=(x_0+x_1,y_0+y_1,z_0+z_1)\) so \[ \large a(x_0+x_1)+b(y_0+y_1)+c(z_0+z_1)= \] \[ \large =(ax_0+by_0+cz_0)+(ax_1+by_1+cz_1)=0+0=0 \] (ii) \(\alpha v=(\alpha x_1,\alpha y_1,\alpha z_1)\) so \[ \large a(\alpha x_1)+b(\alpha y_1)+c(\alpha z_1)=\alpha(ax_1+by_1+cz_1= \alpha0=0 \] this proves that D is a subspace of R^3.
For the converse. Let's suppose that \[ \large D=\{(x,y,z):ax+by+cz=d\} \] is a subspace of R^3. then the zero vector MUST be in D so \[ \large 0=a0+b0+c0=d \]
I do understand that for sure. but how did you know to make d=0, or how do you know it doesn't work for any non zero number?
ohhh okay.
therefore D is a subspace if and only if d=0
so that same idea applies if the question is ax+by=c, for scalars a, b, c ... subspace of R(2) right
yes
It seems like the book is trying to trick me... why give one question stating for ax+by=c, and then the question right after that be ax+by+cz=d, when c has to be 0 for the first and d has to be 0 for the second. Its the same question isnt it, or is there more to it that I am missing.
yes it is. the book is trying to stress that for a subset of a given to be a subspace (i.e., a space on its own) the zero vector (which is unique) must also belong to the subset in question.
that's what u r supposed to get from this two exercises.
*given space*
i hope i was helpful gotta go
it most definitely was... i hope to see you around again, you have been very helpful. thank you.

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