Matrix Transformations: One-to-One and Onto

The video tutorial explains the concepts of one-to-one and onto matrix transformations. It defines one-to-one transformations as those where each output vector has at most one input vector, and onto transformations as those where each output vector has at least one input vector. The tutorial also discusses the properties of these transformations, such as linear independence and spanning, and provides examples using matrices to illustrate these concepts. The video concludes by explaining how to determine if a transformation is one-to-one or onto using matrix pivots.

What is a key characteristic of a one-to-one transformation?

• Every output vector has at least one input vector.

• Every input vector has at most one output vector.

• Every output vector has at most one input vector.

• Every input vector has at least one output vector.

In an onto transformation, what is true about the output vectors?

• They have exactly one input vector.

• They have no input vectors.

• They have at least one input vector.

• They have at most one input vector.

What condition must be met for a transformation to be one-to-one?

• The transformation matrix must have a pivot in every column.

• The transformation matrix must have a pivot in every row.

• The columns of the transformation matrix must be linearly independent.

• The columns of the transformation matrix must span RM.

What is required for a transformation to be onto?

• The transformation matrix must have a pivot in every column.

• The transformation matrix must have a pivot in every row.

• The range of T must have dimension N.

• The columns of the transformation matrix must be linearly independent.

If a transformation matrix has a pivot in every column, what can be concluded?

• The transformation is onto.

• The transformation is both one-to-one and onto.

• The transformation is neither one-to-one nor onto.

• The transformation is one-to-one.

What does it mean if a transformation is both one-to-one and onto?

• The transformation is only one-to-one.

• The transformation is only onto.

• The transformation is neither one-to-one nor onto.

• The transformation is an isomorphism.

In the example of Matrix A, why is it neither one-to-one nor onto?

• It has a pivot in every column but not in every row.

• It has a pivot in every column and every row.

• It lacks a pivot in every column and every row.

• It has a pivot in every row but not in every column.

For Matrix B, why is it considered one-to-one but not onto?

• It lacks a pivot in every column and every row.

• It has a pivot in every column but not in every row.

• It has a pivot in every row but not in every column.

• It has a pivot in every column and every row.

Why is Matrix C considered onto but not one-to-one?

• It has a pivot in every row but not in every column.

• It has a pivot in every column but not in every row.

• It lacks a pivot in every column and every row.

• It has a pivot in every column and every row.

What makes Matrix D both one-to-one and onto?

• It has a pivot in every column but not in every row.

• It lacks a pivot in every column and every row.

• It has a pivot in every row but not in every column.

• It has a pivot in every column and every row.