By Quillen D.G.
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Extra info for Formal properties of over-determined systems of linear partial differential equations [PhD Thesis]
Sample text
Vk ∈ V is the set of all linear combinations of these vectors, denoted by Span(v1 , v2 , . . , vk ). If S is a (finite or infinite) set of vectors in V, then the span of S, denoted by Span(S), is the set of all linear combinations of vectors in S. If V = Span(S), then S spans the vector space V . A (finite or infinite) set of vectors S in V is linearly independent if the only linear combination of distinct vectors in S that produces the zero vector is a trivial linear combination. That is, if vi are distinct vectors in S and c 1 v1 + c 2 v2 + · · · + c k vk = 0, then c 1 = c 2 = · · · = c k = 0.
If A is invertible, then A−1 is also upper (lower) triangular, and the diagonal entries of A−1 are the reciprocals of those of A. In particular, if L is a unit upper (lower) triangular matrix, then L −1 is also a unit upper (lower) triangular matrix. 1-13 Vectors, Matrices, and Systems of Linear Equations 12. , when As and At are defined for integers s and t, then As At = As +t , (As )t = Ast . Examples: 1. For any n, the identity matrix In is invertible and is its own inverse. If P is a permutation matrix, it is invertible and P −1 = P T .
That is, for each x ∈ V there is a unique set of scalars c 1 , c 2 , . . , c p such that x = c 1 b1 + c 2 b2 + · · · + c p b p . Examples: 1. In R2 , 1 0 and are linearly independent, and they span R2 . So they form a basis for R2 and −1 3 dim(R2 ) = 2. 2. In F [x], the set {1, x, x 2 , . . , x n } is a basis for F [x; n] for any n ∈ N. The infinite set {1, x, x 2 , x 3 , . } is a basis for F [x], meaning dim(F [x]) = ∞. 3. The set of m × n matrices E ij having a 1 in the i, j -entry and zeros everywhere else forms a basis for F m×n .