Understanding One-to-One Transformations

This video tutorial explains one-to-one transformations, also known as injective transformations, in linear algebra. It covers the definition, properties, and equivalent conditions for a transformation to be one-to-one. The video also discusses non-one-to-one transformations, providing examples and conditions under which they occur. A theorem relating matrix transformations to one-to-one properties is presented, along with methods to determine if a matrix transformation is one-to-one. The tutorial concludes with demonstrations of matrix transformations, illustrating the concepts discussed.

What is another term used for a one-to-one transformation?

• Surjective

• Bijective

• Injective

• Projective

In a one-to-one transformation, how many solutions can the equation T(x) = b have?

• Zero or one

• Infinite

• Exactly two

• At least one

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

• No input has an output

• Different inputs have different outputs

• Every input has multiple outputs

• Different inputs can have the same output

What condition indicates that a transformation is not one-to-one?

• Every output has multiple inputs

• No input has an output

• Multiple inputs have the same output

• Every input has a unique output

Which of the following is equivalent to a transformation being one-to-one?

• The matrix has no pivots

• The columns of the matrix are linearly dependent

• The kernel contains only the zero vector

• The range of T has dimension less than n

What does it mean if a matrix has a pivot in every column?

• The transformation is not one-to-one

• The matrix is not invertible

• The transformation is one-to-one

• The matrix is singular

What is the first step in determining if a matrix transformation is one-to-one?

• Check if the matrix is square

• Find the inverse of the matrix

• Write the matrix in row echelon form

• Calculate the determinant

In the demonstration, what indicates that the transformation is not one-to-one?

• No output for any input

• Infinite outputs for a single input

• Same output for different inputs

• Different outputs for the same input

What happens when the output vector is changed to the zero vector in a non-one-to-one transformation?

• There is no solution

• The solution is unique

• There are multiple solutions

• There is only the trivial solution

What does the second matrix transformation demonstrate?

• A transformation with no outputs

• A one-to-one transformation

• A transformation projecting onto a plane

• A transformation with infinite inputs