1. Linear Transformation, Null Spaces and Ranges

2. Linear Transformations
Definition:

3. Linear Transformations
Note:

4. Linear Transformations
Example:

5. Linear Transformations
Example:

6. Linear Transformations
Example:

7. Linear Transformations
Example:

8. Linear Transformations
Example:

9. Linear Transformations
Example:

10. The Null Space and Range
Definition T T V W V W O
N(T )
R(T )

11. The Null Space and Range
Example:

12. The Null Space and Range
Example:

13. The Null Space and Range
Example:

14. Conclusion Let T be linear transformation from Rn to Rm.
The null space N(T) is a subspace of Rn.
The range R(T) is a subspace of Rm.
dim(N(T)) + dim(R(T)) = n.
Let A be an m×n matrix
The rank of A is the number of leading 1s in the reduced row echelon form of A.
rank(A) + nullity(A) = n.
An n×n matrix A is nonsingular if and only if rank(A) = n and nullity(A) = 0.