here
... in w1 . Thus, in this case, W1 ⊆ W2 . It follows that either W1 ⊆ W2 or W2 ⊆ W1 . Now, we must prove the converse. We wish to show that, given either W1 ⊆ W2 or W2 ⊆ W1 , we have that W1 ∪ W2 is a subspace of V . Choose a scalar c ∈ F and a vector v ∈ W1 ∪ W2 . Then v is an element of W1 or W2 . But ...
... in w1 . Thus, in this case, W1 ⊆ W2 . It follows that either W1 ⊆ W2 or W2 ⊆ W1 . Now, we must prove the converse. We wish to show that, given either W1 ⊆ W2 or W2 ⊆ W1 , we have that W1 ∪ W2 is a subspace of V . Choose a scalar c ∈ F and a vector v ∈ W1 ∪ W2 . Then v is an element of W1 or W2 . But ...
LECTURE 2 Defintion. A subset W of a vector space V is a subspace if
... closed under linear combinations. Much of today’s class will focus on properties of subsets and subspaces detected by various conditions on linear combinations. Theorem. If W is a subspace of V , then W is a vector space over F with operations coming from those of V . In particular, since all of tho ...
... closed under linear combinations. Much of today’s class will focus on properties of subsets and subspaces detected by various conditions on linear combinations. Theorem. If W is a subspace of V , then W is a vector space over F with operations coming from those of V . In particular, since all of tho ...
Review of Linear Independence Theorems
... Definition. A basis for a vector space V is a list (v1 , . . . , vn ) which is linearly independent and spans V . Corollary of Replacement Theorem. If V is a vector space over a field F , Any two finite bases for V have the same number of elements. The proof is written out in Practice Quiz D. Defini ...
... Definition. A basis for a vector space V is a list (v1 , . . . , vn ) which is linearly independent and spans V . Corollary of Replacement Theorem. If V is a vector space over a field F , Any two finite bases for V have the same number of elements. The proof is written out in Practice Quiz D. Defini ...
Linear Algebra Basics A vector space (or, linear space) is an
... (b) Describe all vectors in the span of {(1, 0, 1), (0, 0, 1)}. (Draw xyz-axes and try to visualize span{(1, 0, 1), (0, 0, 1)}.) (c) If {v1 , . . . , vn } is a linearly dependent set of vectors then show that, for some index k, vk must be a linear combination of v1 , . . . , vk−1 , vk+1 , . . . , vn ...
... (b) Describe all vectors in the span of {(1, 0, 1), (0, 0, 1)}. (Draw xyz-axes and try to visualize span{(1, 0, 1), (0, 0, 1)}.) (c) If {v1 , . . . , vn } is a linearly dependent set of vectors then show that, for some index k, vk must be a linear combination of v1 , . . . , vk−1 , vk+1 , . . . , vn ...
Dated 1/22/01
... by setting the parameters n and k equal to each other or nearly so, and whether there are other natural bases for F . ...
... by setting the parameters n and k equal to each other or nearly so, and whether there are other natural bases for F . ...
