One-to-One and Onto Transformations

The video tutorial explains the concepts of one-to-one and onto transformations. It defines one-to-one transformations as those where each vector in the output space has at most one corresponding input vector, while onto transformations require each vector in the output space to have at least one corresponding input vector. The tutorial analyzes three mappings: the first is one-to-one but not onto, the second is onto but not one-to-one, and the third is both one-to-one and onto.

What is the key difference between one-to-one and onto transformations?

• One-to-one transformations have at most one input for each output.

• Onto transformations have at most one input for each output.

• One-to-one transformations have at least one input for each output.

• Onto transformations have no inputs for some outputs.

In a one-to-one transformation, what is the maximum number of inputs a vector in Rm can have?

• Three

• Two

• One

• Zero

Why is the first mapping not considered onto?

• Because it has less than one input for each output.

• Because it has no input for at least one output.

• Because it has more than one input for some outputs.

• Because it has exactly one input for each output.

Which of the following is true for a one-to-one transformation?

• Every vector in Rm has at least one input in Rn.

• Some vectors in Rm have no inputs in Rn.

• Every vector in Rm has more than one input in Rn.

• Every vector in Rm has at most one input in Rn.

What condition disqualifies the second mapping from being one-to-one?

• It has less than one input for each output.

• It has more than one input for some outputs.

• It has exactly one input for each output.

• It has no inputs for some outputs.

What makes the second mapping onto?

• Every vector in Rm has at least one input in Rn.

• Every vector in Rm has at most one input in Rn.

• Some vectors in Rm have no inputs in Rn.

• Some vectors in Rm have more than one input in Rn.

What does it mean for a transformation to be onto?

• Some vectors in Rm have more than one input in Rn.

• Every vector in Rm has at most one input in Rn.

• Every vector in Rm has at least one input in Rn.

• Some vectors in Rm have no inputs in Rn.

In the third mapping, what characteristic makes it both one-to-one and onto?

• Each vector in Rm has exactly one corresponding input.

• Each vector in Rm has more than one corresponding input.

• Each vector in Rm has less than one corresponding input.

• Each vector in Rm has no corresponding input.

What is the requirement for a transformation to be both one-to-one and onto?

• Each vector in Rm has less than one corresponding input.

• Each vector in Rm has exactly one corresponding input.

• Each vector in Rm has more than one corresponding input.

• Each vector in Rm has no corresponding input.

Which mapping is both one-to-one and onto?

• None of the mappings

• Third mapping

• Second mapping

• First mapping