Algebraic and Differential Methods for Nonlinear Control Theory by Rafael Martínez-Guerra & Oscar Martínez-Fuentes & Juan Javier Montesinos-García

Author:Rafael Martínez-Guerra & Oscar Martínez-Fuentes & Juan Javier Montesinos-García
Language: eng
Format: epub
ISBN: 9783030120252
Publisher: Springer International Publishing

Proof

Suppose that is a linear combination of the other vectors:

Adding to both sides:

With , then the vectors are linearly dependent. Suppose that the vectors are linearly dependent:

Having leads to the solution:

So is a linear combination of the other vectors.

Proposition 6.5

Suppose generates V and is linearly independent, then .

Proof

Since spans V then is linearly dependent and spans V, one of the vectors of is a linear combination of the preceding vectors, but it cannot be hence it must be any of the other denoted , thus if is removed from the set it is left: , doing the same operation to a vector makes which is linearly dependent and spans V. One of the vectors in this last set is a linear combination of the preceding vectors denoted , and removing it from the previous set yields , if this is done again with a new vector and so on, and if leads to the spanning set of the required form , Consider that is not possible, if it where after n operations it will produce which implies that is a linear combination of contradicting the hypothesis that is linearly independent.

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