- We begin this chapter by considering linear combinations of lists of vectors. This leads us to the crucial concept of linear independence. The linear dependence lemma will become one of our most useful tools. - A list of vectors in a vector space that is small enough to be linearly independent and big enough so the linear combinations of the list fill up the vector space is called a **basis** of the vector space. - every basis of a vector space has the same length, which will allow us to define the dimension of a vector space. - This chapter ends with a formula for the dimension of the sum of two subspaces. - • 𝐅 denotes 𝐑 or 𝐂. • 𝑉 denotes a vector space over 𝐅. ## 2A Span and Linear Independence - elements of 𝐅𝑛 $F^n$ (2, −7, 8) ∈ 𝐅3 - lists of vectors (which may be elements of 𝐅𝑛 or of other vector spaces - 2.2: A linear combination of a list 𝑣1, …, 𝑣𝑚 of vectors in 𝑉 is a vector of the form $𝑎_1𝑣_1 + ⋯ + 𝑎_𝑚𝑣_𝑚, where 𝑎_1, …, 𝑎_𝑚 ∈ 𝐅$ - Reform this into a vertical representation: - (17, -4,5) ne $a_1 (2,1,-3) + a_2(1, -2, 4)$ - Def 2.4: span - The set of all linear combinations of a list of vectors 𝑣1, …, 𝑣𝑚 in 𝑉 is called the span of 𝑣1, …, 𝑣𝑚, denoted by span(𝑣1, …, 𝑣𝑚). - In other words, - $ span(𝑣_1, …, 𝑣_𝑚) = {𝑎_1𝑣_1 + ⋯ + 𝑎_𝑚𝑣_𝑚 ∶ 𝑎_1, …, 𝑎_𝑚 ∈ 𝐅}. The span of the empty list ( ) is defined to be {0}. $ - $ \text{span}(v_1, \ldots, v_m) = \{ a_1 v_1 + \ldots + a_m v_m : a_1, \ldots, a_m \in \mathbb{F} \}. \text{ The span of the empty list ( ) is defined to be } \{0\}. $ - Th 2.6: The span of a list of vectors in 𝑉 is the smallest subspace of 𝑉 containing all vectors in the list - How to show "smallest": Two steps: show span is a subspace: - need additive identity: 0 = 0𝑣1 + ⋯ + 0𝑣𝑚. - need closed under addition - need closed under scalar multiplication - showing smallest: 1; each v_k is a linear combination of all v; a_k = 1, a_other =0