testme said:No, we haven't really been taught how to do anything with matrices, except maybe adding matrices or multiplying matrices by a scalar.
When a set of vectors spans Rn, it means that every vector in the n-dimensional space can be written as a linear combination of the given set of vectors. In other words, the set of vectors can reach every point in the n-dimensional space.
To determine if a set of vectors spans Rn, you can use the method of Gaussian elimination. By setting up a matrix with the given vectors as columns and applying row operations, you can check if the matrix has a pivot in every row. If it does, then the set of vectors spans Rn. Alternatively, you can also check if the dimension of the span of the vectors is equal to Rn.
A set of vectors spanning Rn is significant because it forms a basis for the n-dimensional space. This means that any vector in Rn can be uniquely represented as a linear combination of the set of vectors. Additionally, it allows us to solve systems of linear equations and perform other important calculations in the n-dimensional space.
No, a set of vectors must contain at least n vectors to span Rn. This is because a vector space with n dimensions requires at least n linearly independent vectors to span it. If a set contains fewer than n vectors, then it will not be able to reach every point in Rn.
No, a set of vectors must be linearly independent to span Rn. If the vectors are not linearly independent, then they are redundant and can be written as a linear combination of the other vectors in the set. This means that the set will not have enough unique vectors to reach every point in Rn.