Please solve these question Given vector v1, v2, ..., vp in Rn and given scalars
ID: 2961762 • Letter: P
Question
Please solve these question
Given vector v1, v2, ..., vp in Rn and given scalars c1, c2, ... ,cp, the vector y defined by y = c1 v1 + + cp vp is called a linear combination of v1, v2, ..., vp with weights c1, c2, ..., cp. Let V be a set of vectors. If every vector in V can be written as a linear combination of v1, v2, ... , vn, we say that V is spanned or generated by v1, v2, ..., vn and call the set of vectors {v1, v2, ..., vn} a spanning set for V. In this case, we also say that {v1, v2, ... , vn} spans or generates V. Give an example of a set of vectors that spans R2. Give an example of a set of vectors that does not span R2. How does span relate to the matrix equation? How does span relate to linear combination? How does span relate to pivots? Do {[0 1 5],[1 2 8]} span R3? Justify your answer. Do {[0 1 5], [1 2 8], [4 -1 0]} span R3? Justify your answer.Explanation / Answer
- two vectors that span R2 are <1,0> and <0,1>
- two vectors that do not span R2 are <1,0> and <2, 0>
- span is related to matrix equation because if the columns of an n x n matrix A are a spanning set, then in the equation Ax=B, there is only one unique solution for x for every unique n x 1 matrix B.
- span is related to linear combination because if a set of vectors SPANS a set, such as V, then every vector in V can be written as a linear combination of the vectors in the span.
-span is related to pivots because if the columns of an n x n matrix A are a spanning set, then there are exactly n pivots.
-those vectors do not span R3 because for a set of vectors to span R3, there must be at least 3 vectors in the spanning set.
-yes, these vectors span R3, because if you row reduce the matrix with the vectors as columns:
[0 1 4]
[1 2 -1]
[5 8 0]
you get the 3 x 3 identity matrix.