Confused with working out the span of a set of vectors in R^2

Click For Summary
SUMMARY

The discussion centers on the span of the set S, defined as all vectors x=(x₁, x₂) in ℝ² where x₁=1. The conclusion is that span S equals ℝ², as any vector in ℝ² can be expressed as a linear combination of vectors in S. The confusion arises from the conditions where x₁ is both equal to 1 and 0, which are clarified as separate cases to demonstrate that all points in ℝ² can be represented using linear combinations of vectors from S.

PREREQUISITES
  • Understanding of vector spaces and spans in linear algebra
  • Familiarity with linear combinations and their properties
  • Basic knowledge of the real number system ℝ
  • Proficiency in mathematical notation and expressions
NEXT STEPS
  • Study the concept of vector spans in linear algebra
  • Learn about linear combinations and their applications
  • Explore the properties of vector spaces in ℝ²
  • Investigate cases of linear dependence and independence of vectors
USEFUL FOR

Students and educators in mathematics, particularly those focusing on linear algebra, as well as anyone seeking to deepen their understanding of vector spaces and spans in ℝ².

gilabert1985
Messages
7
Reaction score
0
Hi everyone!

I have the following problem which I don't understand... It is already solved, but there are three questions I have regarding it.

The problem says: "Let S be the set of all vectors x=(x_{1}, x_{2}) in \Re^{2} such that x_{1}=1. What is the span of S?"

And here is the answer that has me so confused...

"span S = \Re^{2} because (x_{1}, x_{2})=x_{1}(1, x^{-1}_{1}x_{2} when x_{1}\neq0
and (x_{1}, x_{2})=(1, 0)-(1, -x_{2}) when x_{1}=0."

But I don't understand the first line... why does it say when x_{1}\neq0 if x_{1} is supposed to be equal to 1?

And in the second line, the same... why is x_{1}=0?

So yeah, I understand they are linear combinations and all that, but for the condition given (x_{1}=1), I don't understand how this answer satisfies it.
 
Physics news on Phys.org
The points (x1,x2) in the lines that are confusing you are not points in S, but points in R2, which it then shows how to represent as linear combinations of points in S
 
Since you can't rewrite ##x_2## as ##x_1 x_1{}^{-1}x_2## when ##x_1=0##, you have to consider the case ##x_1=0## separately.
 

Similar threads

Undergrad Proving inequality about dimension of quotient vector spaces
  • · Replies 25 ·
Replies
25
Views
4K
How should I find ## x_{2}(t) ## for this nonlinear integro-differential equation?
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad How are the following three definitions subtly different?
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad Help solving a System of Linear Equations
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Why is there a Matrix A that satisfies F(x,y)=<Ax,y>?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Question on subspaces and spans of vector spaces
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad A very basic question about bra-kets
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Matrix for transforming vector components under rotation
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad I want to expand a Gaussian wavepacket in terms of sines
  • · Replies 2 ·
Replies
2
Views
2K
MHB If a subset T of S is dependent, then S itself dependent
  • · Replies 3 ·
Replies
3
Views
2K