![]() Also, I am self-studying so forums like these are my teachers. Sorry for the long post, I have just been grappling with this for a while so I wanted to clarify. Theorem: If S is any subset of V, the span of S is the smallest linear subspace of V containing S. ![]() Should this not mean that the $span(S)$ is $V$ because of the equality? I can see that subset $S$ could be the smallest part because we are only taking the elements that can span $V$ and that will make sense, but $span(S)$ is supposed to be a set of linear combination and therefore contains every thing that is in $V$. empty set is the set containing just the zero vector. Regardless, where I am confused is that the theorem states that $span(S)$ is the smallest part of $V$, but how can it be the smallest if we are saying that $span(S) = V$ in the definition of the span of a set. The reason why I posted my understanding of the above definitions is so that if I am missing something perhaps someone will point it out to me so I can bridge the gap. Question: Theorem 4.7 is where I am confused. c1 is equal to c2, is equal to all of these. Now, the definition of linear independence meant that the only solution to c1, v1, plus c2, v2 plus all the way to cn, vn, that the only solution to this equally the 0 vector- maybe I should put a little vector sign up there- is when all of these terms are equal to 0. Then a subspace U V is called an invariant subspace under T if. Let V be a finite-dimensional vector space over F with dim ( V) 1, and let T L ( V, V) be an operator in V. ![]() Moreover, $span(S)$ is the smallest subspace of $V$ that contains $S$, in the sense that every other subspace of $V$ that contains $S$ must contain $span(S)$. If you take all of the possibilities of these and you put all of those vectors into a set, that is the span and that's what we're defining the subspace v as. To begin our study, we will look at subspaces U of V that have special properties under an operator T in L ( V, V). The dimension of an affine space is defined as the dimension of the vector space of its translations. Linear spaces are defined in a formal and very general way by enumerating the properties that the two algebraic operations performed on the elements of the. ![]() Linear subspaces, in contrast, always contain the origin of the vector space. I am confused between the definition and the theorem.ĭefinition of Spanning Set of a Vector Space: Let $S = \$ is a set of vectors in a vector space $V$. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. I need a bit of clarification in regards to the spanning set. ![]()
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