# Determine whether the span of the column vectors of the given is in .?

• mottov2
In summary, the question is asking whether the span of the column vectors of the given matrix, which is in R4, is the entire R4 space. This is related to the concept of basis in a vector space, where a set of vectors must be independent, span the space, and have the same dimension as the space. In this case, the matrix will be invertible if the vectors are independent, and therefore the span will be all of R4. There are no examples given in the textbook for this type of question, so further explanation or clarification may be needed.
mottov2
Determine whether the span of the column vectors of the given is in...?

## Homework Statement

determine whether the span of the column vectors of the given matrix is in euclidean space R=4

1 0 1 -1
0 -1 -3 4
1 0 -1 2
-3 0 0 -1

this question is under the inverse of square matrix section of my textbook. Unfortunately it dosent show any examples for this kind of question so I am really clueless.

whats the connection between span and inverse matrices? How can i prove if span of some vectors is in some euclidean space?

Have you given the exact wording of the problem? The phrase "in euclidean space R4" seems strange to me- each of the columns has four numbers and so is in R4 and so their span must be "in" R4 (a subspace of R4). I suspect the question is not whether the span is in R4 (which is trivially true) but whether or not the span is all of R4.

A "basis" for a vector space has three properties:
1) the vectors are independent
2) the vectors span the space
3) the number of vectors in the set is that same as the dimension of the space.

And if any two of those are true, the third is also,

Of course, R4 has dimension 4 and there are 4 columns in that matrix. The vectors will span R4 if and only if the vectors are independent. In that case, it will also be true that the matrix is invertible.

## What is the span of a set of column vectors?

The span of a set of column vectors is the set of all possible linear combinations of those vectors. It represents the entire space that can be reached by scaling and adding the given vectors.

## How do you determine if a span is in a given space?

To determine if a span is in a given space, you can check if the vectors in the span are linearly independent. If they are, then the span is in the given space. Another way is to check if the span contains the zero vector, as it must be in any vector space.

## What is the importance of determining if a span is in a given space?

Determining if a span is in a given space can help in understanding the relationship between different vectors and their span. It also helps in identifying if a set of vectors can be used to form a basis for the given space.

## Can a span be in multiple spaces?

Yes, a span can be in multiple spaces. This can happen if the vectors in the span are linearly independent and can form a basis for multiple spaces.

## How can we visualize the span of a set of column vectors?

One way to visualize the span of a set of column vectors is to plot them in a coordinate system. The span will then represent a plane, line or point depending on the dimension of the vectors. Another way is to use geometric transformations to represent the span in a 3D space.

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