# Relationship of an invertible matrix in spanning set and linear independence

• ichigo444
In summary, a matrix is invertible if and only if its columns (or rows) are linearly independent in Rn, and they span Rn.
ichigo444
What could we say if a matrix is invertible? Could we say that it can span and is linearly independent?

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.

## 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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