Real subspace and not real subspace

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In summary: The Attempt at a SolutionWhat are x and y supposed to be? vectors, components of vectors?The first problem is to determine 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.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...Your 2nd problem is to determine whether the set of 2 x 2 matrices is a subspace
  • #1
xiaobai5883
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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

 
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  • #2
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?...
 
  • #3
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
[tex] (x,y) \in \Re^2 : |x| = |y| [/tex]
then can you think what this rerpresents in [tex]\Re^2[/tex]? 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...
 
  • #4
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
 
  • #5
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.
 
  • #6
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...
 
  • #7
Your 2nd problem is to determine whether the set of 2 x 2 matrices
[tex]
\left[ \begin{array}{ c c }
a & b \\
c & d
\end{array}
\right][/tex]
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:
Code:
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,
[tex]
\left[ \begin{array}{ c }
a \\
b \\
c \\
d
\end{array}
\right] [/tex]=
[tex]d\left[ \begin{array}{ c }
-1 \\
0 \\
0 \\
1
\end{array}
\right] [/tex]
+
[tex]b \left[ \begin{array}{ c }
0 \\
1 \\
0 \\
0
\end{array}
\right][/tex]
+
[tex]c \left[ \begin{array}{ c }
0 \\
0 \\
1 \\
0
\end{array}
\right]
[/tex]
 
Last edited:
  • #8
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...
^^
 
  • #9
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??
 
Last edited:
  • #10
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
 
  • #11
xiaobai5883 said:
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.
 
  • #12
wow...
thanks..
 

1. What is a real subspace?

A real subspace is a subset of a vector space that contains all linear combinations of its vectors and is closed under scalar multiplication. It can also be defined as a subspace that only contains real numbers as its scalars.

2. How is a real subspace different from a non-real subspace?

A real subspace only contains real numbers as its scalars, while a non-real subspace can contain complex numbers as its scalars.

3. Can a complex vector space be considered a real subspace?

Yes, a complex vector space can be considered a real subspace if all of its vectors have real components and only real numbers are used as scalars.

4. What are some examples of real subspaces?

Examples of real subspaces include the set of all real-valued polynomials, the set of all real-valued functions, and the set of all real-valued matrices.

5. How are real subspaces used in practical applications?

Real subspaces are used in various fields of science and engineering, such as physics, computer science, and economics, to model and solve real-world problems. They are also used in linear algebra and functional analysis to study vector spaces and their properties.

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