Linear Algebra (Strang), Question/doubt. n columns lie in plane

[paleywiener]

Hey, anyone (lurking) remember any Linear Algebra? I've been reading
Gilbert Strang: Linear Algebra and Its Applications (4E).

Question 1. Chapter 1(pg 9):
"If the n planes have no point in common, or infinitely many points,
then the n columns lie in the same plane".

Could someone explain this to me?? Basically if n columns lie in the
same plane, AND if some of them are independent of the other (basis
vectors for that space)
then the other vectors(columns) are linear combinations of the basis
vectors. BUT, how does this translate to n planes not having a common
point etc..??

Question 2. Chapter 1(pg 8):
"Now the third equation is the sum of the first two. In that case the
three planes have a whole line in common."

How? u+v+w=2; 2u+3w=5; 3u+v+4w=7; Is there some kind of proof for the
sum of 2 eq's = 3rd implying that they intersect and have a line
common??

Q3.
Could someone recommend a decent book for simple Plane geometry.
Everyone assumes that I know what a plane is but I don't, and
multidimensional planes are even
harder! I understand what a line is: y = Mx + C; Ax + By = C; so if M
or C increases it's easy to visualize what's happening, but planes
(hyperplanes)? 2u + v + w = 5 (or 10) are parallel. I know this
because, I read in another
book (Serge Lange) that two planes are parallel if their normals are
parallel. The normal for both planes here is <2,1,1>, ergo.. . Also,
2u + v = 5 is apparently a plane in 3D - because 0.w can take any sot
of w value.. so the plane actually extends (infinitely) along the w-
dimension.. but stuff like this in Strang confuses me because Strang
assumes you know this stuff.

Q4. Anyone know of a decent free place to ask for Math help?? Usenet??
Any other Math Groups?

[cwldoc]

Post by paleywiener
Hey, anyone (lurking) remember any Linear Algebra?
I've been reading
Gilbert Strang: Linear Algebra and Its Applications
(4E).
"If the n planes have no point in common, or
infinitely many points,
then the n columns lie in the same plane".
Could someone explain this to me?? Basically if n
columns lie in the
same plane, AND if some of them are independent of
the other (basis
vectors for that space)
then the other vectors(columns) are linear
combinations of the basis
vectors. BUT, how does this translate to n planes not
having a common
point etc..??
"Now the third equation is the sum of the first two.
In that case the
three planes have a whole line in common."
How? u+v+w=2; 2u+3w=5; 3u+v+4w=7; Is there some kind
of proof for the
sum of 2 eq's = 3rd implying that they intersect and
have a line
common??

No, there is no such proof. For example, the sum of u+v+w=1 and u+v+w=2 is 2u+2v+2w=3, yet the corresponding planes do not intersect in any points!

Post by paleywiener
Q3.
Could someone recommend a decent book for simple
Plane geometry.
Everyone assumes that I know what a plane is but I
don't, and
multidimensional planes are even
harder! I understand what a line is: y = Mx + C; Ax +
By = C; so if M
or C increases it's easy to visualize what's
happening, but planes
(hyperplanes)? 2u + v + w = 5 (or 10) are parallel. I
know this
because, I read in another
book (Serge Lange) that two planes are parallel if
their normals are
parallel. The normal for both planes here is <2,1,1>,
ergo.. . Also,
2u + v = 5 is apparently a plane in 3D - because 0.w
can take any sot
of w value.. so the plane actually extends
(infinitely) along the w-
dimension.. but stuff like this in Strang confuses me
because Strang
assumes you know this stuff.
Q4. Anyone know of a decent free place to ask for
Math help?? Usenet??
Any other Math Groups?

[The Qurqirish Dragon]

Post by cwldoc

Post by paleywiener
"Now the third equation is the sum of the first two.
In that case the
three planes have a whole line in common."
How? u+v+w=2; 2u+3w=5; 3u+v+4w=7; Is there some kind
of proof for the
sum of 2 eq's = 3rd implying that they intersect and
have a line
common??

No, there is no such proof. For example, the sum of u+v+w=1 and u+v+w=2 is 2u+2v+2w=3, yet the corresponding planes do not intersect in any points!

As with my previous reply, context would have helped here. The
statement in question was made with respect to a SPECIFIC example, not
the general case. The general form of the example (for which the
statement is true) requires that the equations be consistent (the case
of inconsistent equations- which your counterexample uses- is handled
in the preceding section).
As a side note, the section continues to address the "all planes
parallel" case separately, so the intersection being a line (and not a
plane) is accurate here.

[paleywiener]

Hi Qurqirish Dragon, hmm..

My new Eg: 3x + 4y + 5z +7p +5q = 5
2x + 2y + 3z +8p +6q = 6
1x + 1y + 8z +3p +7q = 12

Could you tell me:
1. What is the dimension of the hyperplane represented by
eq: 3x + 4y + 5z +7p +5q = 5
2. What is the dimension of the column vector <3,2,1>
3. What is the dimension of the plane that contains all the column
vectors ASSUMING that they are all linearly independent.
4. What is the dimension of the plane that contains all the column
vectors ASSUMING 2 are dependent vectors.

My own answers would be:
1. 4
2. 3
3. 6
4. 4

[The Qurqirish Dragon]

Post by paleywiener
Hi Qurqirish Dragon, hmm..
My new Eg: 3x + 4y + 5z +7p +5q = 5
           2x + 2y + 3z +8p +6q = 6
           1x + 1y + 8z +3p +7q = 12
1. What is the dimension of the hyperplane represented by
eq: 3x + 4y + 5z +7p +5q = 5
2. What is the dimension of the column vector <3,2,1>
3. What is the dimension of the plane that contains all the column
vectors ASSUMING that they are all linearly independent.
4. What is the dimension of the plane that contains all the column
vectors ASSUMING 2 are dependent vectors.
1. 4

Correct

Post by paleywiener
2. 3

Correct

Post by paleywiener
3. 6

The column vectors CANNOT be linearly independent, since you have more
vectors than dimensions- thus you cannot assume they are (a false
assumption will let you conclude anything). In no case can a set of
vectors span more dimensions than the size of them. In this case, at
most 3. Since there is a set of 3 independent column vectors here,
they span all of the 3-dimensional space. Also, note that the constant
vector (in this example <5,6,12>) is normally not considered when
finding the dimension of the matrix, as it is on the other side of the
equality sign, but if you did include it, that still doesn't change
the dimensionality.

Post by paleywiener
4. 4

See above. at least 3 of the columns there MUST be dependent. (again,
ignoring the right hand side of the equals sign- if you don't than at
least 4 columns must be dependent). A quick check will show you that
any 3 of the 5 (6) columns are independent.

Note also that in order for exactly 2 vectors to be dependent, they
must be multiples of each-other. For more than 2 to be dependent, a
linear combination of all but 1 of them (with non-zero multiples of
each) must equal the final one.
For example:
<1,2,3>; <2.3.4>; <2,4,6> has 2 linearly dependent vectors (the 1st
and 3rd ones), while
<1,2,3>; <2,3,4>; <3,5,7> has 3 linearly dependent vectors. Both span
2 dimensions, however.

[quasi]

On Mon, 16 Aug 2010 08:20:56 -0700 (PDT), paleywiener

Post by paleywiener
Hey, anyone (lurking) remember any Linear Algebra? I've been reading
Gilbert Strang: Linear Algebra and Its Applications (4E).
"If the n planes have no point in common, or infinitely many points,
then the n columns lie in the same plane".
Could someone explain this to me?? Basically if n columns lie in the
same plane, AND if some of them are independent of the other (basis
vectors for that space)
then the other vectors(columns) are linear combinations of the basis
vectors. BUT, how does this translate to n planes not having a common
point etc..??
"Now the third equation is the sum of the first two. In that case the
three planes have a whole line in common."
How? u+v+w=2; 2u+3w=5; 3u+v+4w=7; Is there some kind of proof for the
sum of 2 eq's = 3rd implying that they intersect and have a line
common??
Q3.
Could someone recommend a decent book for simple Plane geometry.
Everyone assumes that I know what a plane is but I don't, and
multidimensional planes are even
harder! I understand what a line is: y = Mx + C; Ax + By = C; so if M
or C increases it's easy to visualize what's happening, but planes
(hyperplanes)? 2u + v + w = 5 (or 10) are parallel. I know this
because, I read in another
book (Serge Lange) that two planes are parallel if their normals are
parallel. The normal for both planes here is <2,1,1>, ergo.. . Also,
2u + v = 5 is apparently a plane in 3D - because 0.w can take any sot
of w value.. so the plane actually extends (infinitely) along the w-
dimension.. but stuff like this in Strang confuses me because Strang
assumes you know this stuff.

I think Lang's "Calculus of Several Variables", Chapter 1 (all of it)
gets right to the heart of the geometry you want to understand.

quasi

[paleywiener]

hey guys, thanks. I'll take a look at "Calculus of several variables".
Any ideas about Q1, Q2?? I've quoted Strang directly..

[The Qurqirish Dragon]

Post by paleywiener
Hey, anyone (lurking) remember any Linear Algebra? I've been reading
Gilbert Strang: Linear Algebra and Its Applications (4E).
"If the n planes have no point in common, or infinitely many points,
then the n columns lie in the same plane".
Could someone explain this to me?? Basically if n columns lie in the
same plane, AND if some of them are independent of the other (basis
vectors for that space)
then the other vectors(columns) are linear combinations of the basis
vectors. BUT, how does this translate to n planes not having a common
point etc..??

It would have helped if you gave the full context of the statement.
Fortunately, I have a copy of this book (3E), and I could check it.

first, remember the definition of an "n plane": it is the (n-1
dimensional hyper-)plane defined by row 1 of the matrix. For example,
in the 3x3 matrix problem:
[1 2 3] [x] = [4]
[4 5 6] [y] = [8]
[7 8 9] [z] = [9]

the "3 planes" are the 2-dimensional planes defined by the equations:
x + 2y + 3z = 4
4x + 5y +6z = 8
7x + 8y +9z = 9

Now, what the statement "If the n planes have no point in common, or
infinitely many points then the n columns lie in the same plane" means
is:
if the intersection of all the {n-1 dimensional} planes is empty, or
contains infinitely many points, then the column-vectors all lie in a
single {n-1 dimensional} plane.

In other words, if there is no solution to the system of equations
associated to the matrix problem, then the matrix is singular, and so
the columns are dependent (and so all the column vectors lie in a n-1
dimensional plane).

Note that the column vectors could lie in a space of smaller dimension
than n-1 as well.