Questions and Answers
Define a vector space V over a field F.
A vector space V over a field F is a set V with two operations: vector addition and scalar multiplication, satisfying specific axioms.
What are the four properties of a vector space V over a field K?
- For any scalar k∈K and 0∈V, k0=0. 2. For 0∈K and any vector u∈V, 0u=0. 3. If ku=0, then k=0 or u=0. 4. For any k∈K and any u∈V, (-k)u=k(-u)=-ku.
If v1, v2,..., vn∈V are vectors in a vector space, a linear combination of them is?
A vector of the form α1v1+α2v2+...+αmv_m where α1,...,αm∈F.
What does it mean for vectors u1, u2,..., um in V to span V?
Signup and view all the answers
What is the (linear) span of v1,..., vm?
Signup and view all the answers
When is W considered a subspace of V?
Signup and view all the answers
W is a subspace of V if the zero vector belongs to W.
Signup and view all the answers
If you add any vector to a spanning set, you are left with a spanning set.
Signup and view all the answers
If one of the vectors in a spanning set can be written as a linear combination of the others, then that spanning set still spans V.
Signup and view all the answers
If the zero vector is in a spanning set, then removing it means the set does not span V.
Signup and view all the answers
What can be said about any linear combination of u1, u2,..., un if they are in the span of v1, v2,..., vn?
Signup and view all the answers
A vector space V is said to be of finite dimension n if?
Signup and view all the answers
What is the dimension of the vector space {0}?
Signup and view all the answers
What does the theorem about span(S) state?
Signup and view all the answers
What do row equivalent matrices have in common?
Signup and view all the answers
What do A and B have in common if they are row equivalent echelon matrices?
Signup and view all the answers
A and B have the same ____ if and only if they have the same nonzero rows.
Signup and view all the answers
Every matrix A is row equivalent to a unique?
Signup and view all the answers
Vectors v1, v2,..., vm in V are linearly dependent if?
Signup and view all the answers
How do you show that a set of vectors v1,...,vm are linearly independent?
Signup and view all the answers
If v1, v2,..., vm are dependent, then any set containing them is also dependent.
Signup and view all the answers
Study Notes
Vector Space and Definitions
- A vector space ( V ) over a field ( F ) includes two operations: vector addition and scalar multiplication.
- For ( V ) to qualify as a vector space, several axioms (A1-A4, M1-M4) must be satisfied by any vectors ( u, v, w ) in ( V ).
Properties of Vector Spaces
- Property 1: For every scalar ( k \in K ), ( k0 = 0 ).
- Property 2: For ( 0 \in K ) and any vector ( u \in V ), ( 0u = 0 ).
- Property 3: If ( ku = 0 ) with ( k \in K ) and ( u \in V ), then either ( k = 0 ) or ( u = 0 ).
- Property 4: For any scalar ( k \in K ) and any vector ( u \in V ), ( (-k)u = k(-u) = -ku ).
Linear Combinations
- A linear combination of vectors ( v_1, v_2, \ldots, v_n ) is expressed as ( \alpha_1v_1 + \alpha_2v_2 + \ldots + \alpha_mv_m ) where ( \alpha_i \in F ).
Spanning Sets
- Vectors ( u_1, u_2, \ldots, u_m ) in vector space ( V ) span ( V ) if every vector ( v ) in ( V ) can be expressed as a linear combination of the vectors ( u_i ).
- The span of vectors ( v_1, \ldots, v_m ) is defined as the set of all linear combinations of these vectors.
Subspaces
- A subset ( W ) of vector space ( V ) is a subspace if it is itself a vector space with respect to the operations of ( V ).
- For ( W ) to be a subspace, it must include the zero vector and be closed under vector addition and scalar multiplication.
Properties of Spanning Sets
- Adding any vector to a spanning set does not change the spanning property.
- If a vector in a spanning set can be represented as a combination of others, removing it still results in a spanning set.
- If zero is part of a spanning set, the set without zero also spans the same space.
Span and Linear Combinations
- If ( u_1, u_2, \ldots, u_n ) are in the span of ( v_1, v_2, \ldots, v_n ), then any linear combination of ( u_i ) is also in the span of ( v_i ).
Dimension of Vector Spaces
- A vector space ( V ) is defined as ( n )-dimensional (written ( \text{dim } V = n )) if it has a basis consisting of ( n ) elements.
- The dimension of the trivial vector space ( {0} ) is zero, represented as ( \text{dim } {0} = 0 ).
Theorems Related to Spanning Sets
- For any subset ( S ) of ( V ), ( \text{span}(S) ) is a subspace that contains ( S ) and is the smallest subspace of ( V ) that contains ( S ).
- Row equivalent matrices share the same row space and have the same number of non-zero rows along with pivot entries in the same positions.
Linear Dependence and Independence
- Vectors ( v_1, v_2, \ldots, v_m ) are considered linearly dependent if there are scalars ( a_i ), not all zero, such that ( a_1v_1 + a_2v_2 + \ldots + a_m v_m = 0 ).
- To prove that vectors ( v_1, \ldots, v_m ) are linearly independent, demonstrate that ( x_1 v_1 + \ldots + x_m v_m = 0 ) implies ( x_1 = 0, \ldots, x_m = 0 ).
- Any set that includes dependent vectors is also dependent.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge of vector spaces and their properties with these flashcards covering essential concepts in linear algebra. Each card provides key definitions and axioms critical to understanding vector spaces. Perfect for students learning linear algebra.
