Real subspace and not real subspace

[xiaobai5883]

Homework Statement

1) Explain why the set W={(x,y)inside dimension 2; |x|=|y|} is not a real subspace

2) Show that the set V={[a b];a+d=0} is a real subspace of dimension 3
{[c d] }

Homework Equations

The Attempt at a Solution

 

[sutupidmath]

What are x and y supposed to be? vectors, components of vectors?

the same for [a b] what is this supposed to mean? a row vector?...

 

[lanedance]

hi xiaobai5883

any ideas how to go approach it? also easier if you provide more info as its a bit hard to work out what you're trying to do...

if the first means
(x,y) \in \Re^2 : |x| = |y|
then can you think what this rerpresents in \Re^2? think lines in a plane...

do you know then know the subspace requirements? will need to show it breaks some of those - closure under addition spring to mind...

 

[xiaobai5883]

sorry i didn't mention well about my quesitons...
lanedance is right at the first question...
i know what subspace requirement but i don't know whether the meaning of subspace and real subspace is the same or not...

my second question should be
[a b]
[c d]
it is a 2*2 matrix... then a+d=0
and need to show the real subspace of R^3

 

[HallsofIvy]

If you are expected to be able to a problem like this, you are expected to know what a "subspace" IS. What is the definition of "subspace"? (Don't worry about the word "real". Here that just means "really is a subspace".)

In the first problem, tell which of the parts of the definition are not true. (Hint: what is zero times any vector?)

For the second problem, show that each of the parts of the definition are true.

 

[xiaobai5883]

HallsofIvy thanks for remind me something back in my mind...
thanks a lot...
my problems are solve...
i totally know what is subspace...
but i don't know how to proof a matrix that 2*2 is a subspace of R^3...

 

[Mark44]

Your 2nd problem is to determine whether the set of 2 x 2 matrices
<br /> \left[ \begin{array}{ c c }<br /> a &amp; b \\<br /> c &amp; d<br /> \end{array} <br /> \right]
where a + d = 0, is a subspace (of dimension 3) of the vector space of 2 x 2 matrices.

There are two parts to this problem:

1. Showing that this set of matrices is a subspace.
2. Finding the dimension of this subspace.

For the first part, show that:

1. The 2 x 2 zero matrix belongs to this set.
2. If M1 and M2 are in this set, then M1 + M2 is also in the set.
3. If M is a matrix in this set, and c is a scalar (a real number), then cM is also in this set.

For the second part you have two pieces of information to work with: the equation a + d = 0, and the fact that the entries of matrices in this set are a, b, c, and d, reading across the rows.

From the equation you are given, you can get four equations:
a = -d
b = b
c = c
d = d

Equivalently, this system is:

a =       -d
b = b
c =    c
d =        d

Another way to look at this system is that the entries on the left side represent your matrix (as a vector), and the right side entries can be thought of as the sum of 3 vectors/matrices.

That is,
<br /> \left[ \begin{array}{ c }<br /> a \\<br /> b \\<br /> c \\<br /> d<br /> \end{array} <br /> \right]=
d\left[ \begin{array}{ c }<br /> -1 \\<br /> 0 \\<br /> 0 \\<br /> 1<br /> \end{array} <br /> \right]
+
b \left[ \begin{array}{ c }<br /> 0 \\<br /> 1 \\<br /> 0 \\<br /> 0<br /> \end{array} <br /> \right]
+
c \left[ \begin{array}{ c }<br /> 0 \\<br /> 0 \\<br /> 1 \\<br /> 0<br /> \end{array} <br /> \right]<br />

 

[xiaobai5883]

Mark44...
Thanks very much for you clear explanation...
i'm understand very well also...
thanks thanks thanks...
have a nice day to everyone that help me...
^^

 

[xiaobai5883]

wait wait...
hold on...
still another question...
how about the finding of the basis for V??
is it i need to let 3 matrix myself and find the basis by that??

 

[Office_Shredder]

The use of 'real' subspace could be a subtle point regarding the use of subfields to generate new vector spaces that have a potentially broader class of subspaces (example: complex numbers are a 1 dimensional vector spaceover C, but a 2 dimensional vector space over R. So R is not a subspace of C over C, but is over R). It's not really relevant to this question though

 

[Mark44]

wait wait...
hold on...
still another question...
how about the finding of the basis for V??
is it i need to let 3 matrix myself and find the basis by that??

Take another look at post 7. I fixed the bad LaTeX tags I had earlier, so you'll be able to see what I had in there that wasn't showing.

 

[xiaobai5883]

wow...
thanks..