Basis and dimension of the solution space

  • Thread starter aleee
  • Start date
  • #1
17
0

Homework Statement


Find (a) a basis for and (b) the dimension of the solution space of the homogeneous system of equations.

x - 2y + z = 0
y - z + w = 0
x - y + w = 0

Homework Equations



The Attempt at a Solution


(a)
[1 -2 1 0] => [1 0 -1 2]
[0 1 -1 1] => [0 1 -1 1]
[1 -1 0 1] => [0 0 0 0]
basis = {<1,0,-1,2>, <0,1,-1,1>}

(b) for b i make
x = z - 2w
y = z - w

would i set w = s and z = t?
if so.
[x] = [ t - 2s] = [1] + [-2]
[y] = [ t - s] = t[1] + s[-1]
[z] = [ t ] = [1] + [0]
[w] = [ s ] = [0] + [1]

so the dimension would be the number of vectors
so dimension = 2?

im uncertain about that dimension part
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
770
I didn't check your arithmetic, but assuming it is OK try writing your solution like this:

[tex]\left ( \begin{array}{c}x\\y\\z\\w\end{array}\right) =
t\left ( \begin{array}{c}1\\1\\1\\0\end{array}\right)
+s\left ( \begin{array}{c}-2\\-1\\0\\1\end{array}\right)[/tex]
and the answer should become apparent.
 
  • #3
vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
15,228
1,830
You found that the solution of the system is given by

[tex]\begin{bmatrix}x\\y\\z\\w\end{bmatrix} = t\begin{bmatrix}1\\1\\1\\0\end{bmatrix}+s\begin{bmatrix}-2\\-1\\0\\1\end{bmatrix}[/tex]

The vectors multiplying t and s are a basis for the solution space. Your answer to (a) is the basis for the row space of the matrix, not the solution space.

The dimension is just the number of vectors in the basis, so in this case, it's 2.
 
  • #4
17
0
oh so for (a) i got it mixed up with the basis for the row space
the correct answer is { <1,1,1,0>, < -2,-1,0,1>}
 

Related Threads on Basis and dimension of the solution space

  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
1
Views
8K
Replies
1
Views
2K
  • Last Post
Replies
10
Views
6K
Replies
2
Views
2K
Replies
5
Views
7K
Replies
1
Views
9K
Replies
8
Views
4K
  • Last Post
Replies
2
Views
6K
  • Last Post
Replies
14
Views
18K
Top