- #1
ichigo444
- 12
- 0
What could we say if a matrix is invertible? Could we say that it can span and is linearly independent?
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.
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.
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.
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.
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.