Understanding Onto Transformations

The video tutorial explains onto transformations, also known as surjective transformations, where every output vector in the codomain has at least one corresponding input vector in the domain. It provides examples of onto and non-onto transformations, illustrating the conditions under which a transformation is onto. The tutorial also covers matrix transformations, detailing how to determine if a matrix transformation is onto by checking for pivots in every row of the matrix in row echelon form. Demonstrations of matrix transformations are provided to reinforce the concepts discussed.

What is another term used for an onto transformation?

• Injective

• Bijective

• Surjective

• Reflective

In an onto transformation, what must be true for every output vector?

• It has more than two corresponding input vectors

• It has no corresponding input vector

• It has at least one corresponding input vector

• It has exactly two corresponding input vectors

What condition indicates a transformation is not onto?

• The range is infinite

• The range is equal to the codomain

• The range is larger than the codomain

• The range is smaller than the codomain

What is the range of a transformation that is not onto?

• Smaller than the codomain

• Larger than the codomain

• Infinite

• Equal to the codomain

What does it mean if a vector in the codomain has no corresponding input vector?

• The transformation is injective

• The transformation is bijective

• The transformation is not onto

• The transformation is onto

Which of the following is equivalent to a transformation being onto?

• The matrix is diagonal

• The matrix has a pivot in every row

• The columns of the matrix span R^n

• The matrix has a pivot in every column

What is the significance of having a pivot in every row of a matrix?

• It indicates the matrix is symmetric

• It indicates the matrix is invertible

• It indicates the transformation is injective

• It indicates the transformation is onto

In the context of matrix transformations, what does the rank of a matrix represent?

• The determinant of the matrix

• The dimension of the range

• The number of rows

• The number of columns

How can you determine if a matrix transformation is onto?

• By checking if the matrix is symmetric

• By checking if the matrix is in column echelon form

• By checking if the matrix is invertible

• By checking if the matrix is in row echelon form with a pivot in every row

What happens if the range of a transformation is a plane in R^3?

• The transformation is onto

• The transformation is injective

• The transformation is not onto

• The transformation is bijective