Can Vectors Span Rm? Exploring the Possibility with Calculus Prerequisites

In summary, the conversation discusses the possibility of a set of n vectors in Rm spanning all of Rm. The question is raised if a set of n vectors can span Rm, and the attempt at a solution suggests that it is possible if there is a pivot point in each column. The conversation also brings up the question of whether one vector can span R2, and asks for an explanation.
  • #1
MustangGt94
9
0
Not sure if this is "beyond" Calculus but the prereq for the class was Calc II so I thought i'd post here.

Homework Statement


Could a set of n vectors in Rm span all of Rm?

Homework Equations



None

The Attempt at a Solution


Hmm really stuck at this one. I think that it should be able to span since the only thing that matters is if you have a pivot point in each column? So even if you have say 3 vectors in R4 it could still span it, since you can have still have a pivot point in each of the column if the vectors allow it?

Thanks for the help.
 
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  • #2
n<m, right? Can one vector span R2? Why or why not?
 

What is a vector?

A vector is a mathematical object that has both magnitude (size) and direction. It is represented by an arrow pointing in the direction of the vector and its length represents its magnitude. Vectors are commonly used in physics and engineering to represent forces, velocities, and other physical quantities.

What is Rm?

Rm is a mathematical notation that represents the m-dimensional Euclidean space. It is used to denote a space with m axes, each representing a different dimension. In other words, Rm is a set of all possible points that can be represented by m real numbers.

Can vectors span Rm?

Yes, vectors can span Rm. This means that a set of vectors in Rm can be used to represent any point in that space. By choosing the right combination of vectors, any point in Rm can be reached.

How many vectors are needed to span Rm?

To span Rm, a set of m linearly independent vectors is needed. This means that none of the vectors in the set can be written as a linear combination of the others. If the set contains fewer than m vectors, it will not be able to span Rm.

What is the difference between spanning and spanning linearly?

Spanning linearly means that the vectors in a set can be combined using only scalar multiplication and addition/subtraction to reach any point in Rm. Spanning without the "linearly" qualifier means that the vectors can be combined using any mathematical operation, including multiplication and division, to reach any point in Rm. In other words, spanning linearly is a more specific requirement than simply spanning.

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