1. 4.3 Linearly Independent Sets; Bases
4 Vector Spaces
4.3 Linearly Independent Sets; Bases

2. REVIEW
Definition
A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars, subject to the ten axioms:

3. REVIEW
Definition
A subspace of a vector space V is a subset H of V that satisfies
The zero vector of V is in H.
H is closed under vector addition.
H is closed under multiplication by scalars.

4. REVIEW Theorem 1 If are in a vector space V,
then Span is a subspace of V.

5. REVIEW Definition The null space of an matrix A, written as Nul A, is
the set of all solutions to Ax=0.
Theorem 2
The null space of an matrix A is a subspace of

6. REVIEW
Definition
The column space of an matrix A, written as Col A, is
the set of all linear combinations of the columns of A.
Note:
Theorem 3
The column space of an matrix A is a subspace of

7. REVIEW
Definition:
A linear transformation T from a vector space V into a vector
space W is a rule assigns to each vector x in V a unique vector
T(x) in W, such that
(i) T(u+v)=T(u)+T(v) for all u, v in the domain of T:
(ii) T(cu)=cT(u) for all u and all scalars c.
If T(x)=Ax for some matrix A, Kernel of T = Nul A
Range of T = Col A.

8. 4.3 Linearly Independent Sets; Bases
Purpose: To study the vectors that span a vector space (or a subspace) as efficiently as possible.

9. Linear Independence
in V is linearly independent
has only the trivial solution.

10. Tips to determine the linear dependence
A set is linearly dependent, if it satisfies one of the following:
A set has two vectors and one is a multiple of the other.
2. A set has two or more vectors and one of the vectors is a
linear combination of the others.
3. A set contains more vectors than the number of entries in
each vector.
A set contains the zero vector.

11. Theorem 4
An indexed set of two or more vectors, with ,
is linearly dependent
if and only if some is a linear combination of the
preceding vectors,
Example:

12. Definition
Let H be a subspace of a vector space V.
An indexed set of vectors in V is a basis for H if
i) is a linearly independent set, and
ii) the subspace spanned by coincides with H; i.e.

13. Examples:
1. Let A be an invertible matrix. Then the columns of A
form a basis for Why?
2. Let be the columns of the identity matrix I.
Then, is called the standard basis for

14. 4. Let
Determine if is a basis for
5. Let
Determine if is a basis for
6. Let
Determine if is a basis for

15. The Spanning Set Theorem
Let be a set in V, and let
If one of the vectors in S, say , is a linear combination
of the remaining vectors in S, then the set formed from S by
removing still spans H.
b. If , some subset of S is a basis for H.

16. Example: Find a basis for Col A, where

17. Example: Find a basis for Col B, where

18. Theorem
The pivot columns of a matrix A form a basis for Col A.