Why is it not possible for the columns of a matrix to span R^2 even if

Click For Summary
SUMMARY

The discussion clarifies that the columns of a matrix can indeed span R² while not spanning R³, as demonstrated by the matrix \begin{bmatrix}1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 0 & 0\end{bmatrix}, which has a rank of 2. The confusion arises from misinterpreting the conditions under which matrix columns can span different dimensional spaces. Specifically, a rank-1 matrix cannot span R², but a rank-2 matrix can span R² without spanning R³.

PREREQUISITES
  • Understanding of matrix rank and its implications
  • Familiarity with linear algebra concepts, particularly spanning sets
  • Knowledge of row-reduction techniques for matrices
  • Basic comprehension of vector spaces, specifically R² and R³
NEXT STEPS
  • Study the concept of matrix rank in detail
  • Learn about spanning sets and their properties in linear algebra
  • Explore row-reduction techniques and their applications
  • Investigate the differences between R² and R³ vector spaces
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to clarify concepts related to matrix theory and vector spaces.

fackert
Messages
3
Reaction score
0
Why is it not possible for the columns of a matrix to span R^2 even if those columns do not span R^3?
 
Physics news on Phys.org


You may end up with a rank-1 matrix after row-reduction.
 


fackert said:
Why is it not possible for the columns of a matrix to span R^2 even if those columns do not span R^3?
Is that really what you meant to ask? There is no "why" for a statement that isn't true to begin with! The columns of the matrix
[tex]\begin{bmatrix}1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & 0 & 0\end{bmatrix}[/tex]
do span R2 but not R3.

(Bacle answered a different question: Why is it possible that the columns of a matrix do not span R2?)
 

Similar threads

Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Existence and Uniqueness of Inverses
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Index notation of vector rotation
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Why Use Gram-Schmidt to Make a Unitary Matrix?
  • · Replies 14 ·
Replies
14
Views
4K
Undergrad Is the Projection Matrix A^2=A Valid in R^4?
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Matrix rank in terms of determinant polynomial
  • · Replies 14 ·
Replies
14
Views
5K
MHB If 2 columns are identical is there infinite solutions
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Question about row space basis and Column space basis
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Cycles from patterns in a permutation matrix
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Dimension and solution to matrix-vector product
  • · Replies 4 ·
Replies
4
Views
3K