Fredrik said:It's easy to show that if a matrix A is invertible, then so is its transpose AT. Do you know any facts about invertible matrices that might help you do this?
Linear independence refers to a set of vectors that cannot be expressed as a linear combination of each other. In other words, no vector in the set can be written as a sum of the other vectors multiplied by some scalar constants.
To determine if a set of vectors is linearly independent, you can use the method of Gaussian elimination or row reduction to put the vectors into a matrix and check for linear dependence. Alternatively, you can also use the determinant of the matrix formed by the vectors - if the determinant is non-zero, the vectors are linearly independent.
Yes, a set of vectors can be linearly dependent in one vector space but linearly independent in another. This is because the concept of linear independence depends on the underlying vector space and the operations defined on it.
A subspace of a vector space is a subset that satisfies all the properties of a vector space. It can contain linearly independent or dependent vectors, but the subspace itself is linearly independent if and only if its basis vectors are linearly independent.
Linear independence is important because it allows us to represent a wide variety of vector spaces in a more efficient and concise manner. It also helps us understand the structure and relationships between different vector spaces and their subspaces.