Relationship of an invertible matrix in spanning set and linear independence

In summary, a matrix is invertible if and only if its columns (or rows) are linearly independent in Rn, and they span Rn.
  • #1
ichigo444
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0
What could we say if a matrix is invertible? Could we say that it can span and is linearly independent?
 
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  • #2
I have no idea what you are talking about! "Span" and "linearly independent" are properties of sets of vectors, not matrices. Are you referring to the columns or rows as a vectors?

If so, then, yes, a matrix is invertible if and only if its columns (equivalently, rows) thought of as vectors in Rn are independent.

But I still don't know what you mean by "can span". Can span what? Any set of vector spans something. It is true that if an n by n matrix is invertible then its columns (equivalently, rows) thought of as vectors in Rn span Rn.
 

Related to Relationship of an invertible matrix in spanning set and linear independence

1. What is an invertible matrix?

An invertible matrix is a square matrix in which every row and every column is linearly independent. This means that the columns and rows cannot be expressed as a linear combination of each other. An invertible matrix is also known as a non-singular matrix.

2. How is an invertible matrix related to spanning sets?

An invertible matrix is related to spanning sets because it can be used to determine whether a set of vectors is a spanning set. If the columns of an invertible matrix span the entire space, then the vectors in the matrix are considered to be a spanning set for that space.

3. Can an invertible matrix be linearly dependent?

No, an invertible matrix cannot be linearly dependent. As mentioned earlier, an invertible matrix is a square matrix in which every row and every column is linearly independent. This means that the columns and rows cannot be expressed as a linear combination of each other, making it impossible for an invertible matrix to be linearly dependent.

4. Is every linearly independent set of vectors also a spanning set?

No, not every linearly independent set of vectors is a spanning set. A set of vectors is considered a spanning set if they can be used to create any vector in a given space through linear combinations. However, a linearly independent set of vectors can still exist within a larger set of vectors that is a spanning set.

5. How can an invertible matrix be used to test for linear independence?

An invertible matrix can be used to test for linear independence by creating a matrix with the given vectors as columns. If the determinant of the matrix is non-zero, then the vectors are linearly independent. This is because a non-zero determinant indicates that the columns are linearly independent, and therefore the vectors are also linearly independent.

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