Linear Transformation, Null Spaces and Ranges

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Presentation transcript:

Linear Transformation, Null Spaces and Ranges

Linear Transformations Definition:

Linear Transformations Note:

Linear Transformations Example:

Linear Transformations Example:

Linear Transformations Example:

Linear Transformations Example:

Linear Transformations Example:

Linear Transformations Example:

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

The Null Space and Range Example:

The Null Space and Range Example:

The Null Space and Range Example:

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.