Connections Between Onto/one To One

Question 1

What does it mean to be onto

Answer 1

The range of the transformation spans the dimension of the output/image. (Range(T)=R^m) and also, there is a preimage for each vector in the vector space of the image (Ax=b is consistent for all b in R^m)

Question 2

What does it mean to be one to one

Answer 2

The only solution to the homogeneous equation is the trivial solution. There are no free variables. (Ax=0 only has trivial solution and Ker(T) ={0n))

Question 3

If each element in the codomain is a linear combo of the standard matrixs columns, is this onto or one to one?

Answer 3

Onto

Question 4

If there’s a pivot in every row what does that mean?

Answer 4

It means that the transformation is onto only if there are as much or more columns than rows (m<=n)

Question 5

If there’s a pivot in each column, what does that mean?

Answer 5

It means the transformation is one to one only if there are as much or more rows than there is columns (n <= m)

Question 6

If the columns of A are LI, then

Answer 6

The transformation is one to one

Question 7

What is a bijection?

Answer 7

When you are onto AND one to one

Question 8

What does it mean to be singular?

Answer 8

To not be invertible

Question 9

If A is invertible then

Answer 9

The matrix is square and has exactly n pivots on each row and coloumn,

Ax=b has a unique solution due to being one to one (x=A^-1b) and also has a solution for every b in R^n as it is onto

The columns of A form a basis for R^n (because by definition, basis is when u are LI and span the entire vector space)

And that’s only true because the transformation is a bijection, it’s both onto and one to one