When a set of vectors spans a set W, it means that every vector in set W can be written as a linear combination of the vectors in the set. In other words, the vectors in the set can be used to create any vector in set W through scalar multiplication and addition.
To find vectors that span a set W, you can use the process of Gaussian elimination. Write the vectors in set W as column vectors in a matrix and perform row operations until the matrix is in reduced row-echelon form. The resulting rows will be the vectors that span set W.
Yes, a set of vectors can still span a set W even if it contains redundant vectors. This is because the redundant vectors can be written as linear combinations of the other vectors in the set, and therefore do not contribute any new information to the span of set W.
Yes, it is possible for a set of vectors to span a set W in multiple ways. This is because there can be many different combinations of the vectors that can be used to create the same vector in set W through scalar multiplication and addition.
Yes, a set of vectors can still span a set W even if the vectors are not all in the same dimension. This is because the vectors can still be used to create linear combinations that result in vectors of any dimension in set W.