1. CHAPTER 4 Vector Spaces Linear combination Sec 4.3 IF
The vector w is called a linear combination of the vectors IF
there exists scalars
such that
Find echelon of:
Determine whether the vector w is a linear combination of the vectors
Determine whether the vector w is a linear combination of the vectors
Determine whether the vector w is a linear combination of the vectors
No sol
consistent
Not linear combination
linear combination

2. CHAPTER 4 Vector Spaces Linear combination Sec 4.3 IF
The vector w is called a linear combination of the vectors IF
there exists scalars
such that
v is a linear combination of u1,u2

3. Linearly dependent vectors
CHAPTER 4 Vector Spaces
Sec 4.3
Linearly dependent vectors
are said to be linearly dependent provided that one of them is a linear combination of the remaining vectors
otherwise, they are linearly independent
{ f, g, h } are linearly dependent
{ f, g, h } are linearly dependent
Determine whether the vector w is a linear combination of the vectors
v is a linear combination of u1,u2
{ u1, u2, v} are linearly dependent

4. Space spanned by vectors
CHAPTER 4 Vector Spaces
Sec 4.3
Space spanned by vectors
W = set of all possible linear combination of the vectors
We say W is spanned by the vectors v1 and v2
{v1, v2} is called the spanning set for W.
THM1:

5. Linearly dependent vectors
CHAPTER 4 Vector Spaces
Sec 4.3
Linearly dependent vectors
are said to be linearly dependent provided that one of them is a linear combination of the remaining vectors
otherwise, they are linearly independent
Linearly independent vectors
are said to be linearly independent provided that the equation:
has only the trivial solution

6. Linearly dependent vectors
CHAPTER 4 Vector Spaces
Sec 4.3
Linearly dependent vectors
We solve:
homog
echelon
Only the trivial solution

7. Linearly dependent vectors
CHAPTER 4 Vector Spaces
Sec 4.3
Linearly dependent vectors
We solve:
homog
WHY?
Inf. sol
Remark

8. Linearly dependent vectors
CHAPTER 4 Vector Spaces
Sec 4.3
Linearly dependent vectors

9. Linearly independent vectors
CHAPTER 4 Vector Spaces
Sec 4.3
Linearly independent vectors
are said to be linearly independent provided that the equation:
has only the trivial solution
Linearly dependent vectors
There exists scalars
not all zeros such that
linearly dependent

10. Linearly dependent vectors
CHAPTER 4 Vector Spaces
Sec 4.3
Linearly dependent vectors
Only the trivial solution
Remark

11. Linearly dependent vectors
CHAPTER 4 Vector Spaces
Sec 4.3
Linearly dependent vectors
There exists scalars
not all zeros such that
linearly dependent
At least one of them is a linear combination of the others
linearly dependent

12. TH2: independence of n vectors in R^n
CHAPTER 4 Vector Spaces
Sec 4.3
TH2: independence of n vectors in R^n
The matrix
linearly independent
has nonzero determinant
Columns of the matrix A
linearly independent
linearly dependent

13. TH2
This is true only for n vectors in R^n

14. NOTE:
Def or th3
??????
Use determinant

15. Theorem7:(p193) A All statements are equivalent row equivalent Ax = b
Columns of
are linearly independent
row equivalent
Ax = b
Every n-vector b
has unique sol
is a product of elementary matrices
Ax = b
Every n-vector b
is consistent
Ax = 0
The system
has only the trivial sol
nonsingular
All statements are equivalent

16. TH3: Fewer than n vectors in R^n
CHAPTER 4 Vector Spaces
Sec 4.3
TH3: Fewer than n vectors in R^n
The matrix
nxk
Some kxk submatrix of A has nonzero determinant
linearly independent

17. TH3: Fewer than n vectors in R^n
CHAPTER 4 Vector Spaces
Sec 4.3
TH3: Fewer than n vectors in R^n
The matrix
nxk
Some kxk submatrix of A has nonzero determinant
linearly independent
Size???
How many submatrix??