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Do ups use a lot of power?

## How do you know if a linear transformation is onto a one-to-one?

If there is a pivot in each column of the matrix, then the columns of the matrix are linearly indepen- dent, hence the linear transformation is **one-to-one**; if there is a pivot in each row of the matrix, then the columns of A span the codomain Rm, hence the linear transformation is onto.

## How do you know if a function is one-to-one?

If the graph of a function f is known, it is easy to determine if the function is 1 -to- 1 . **Use the Horizontal Line Test**. If no horizontal line intersects the graph of the function f in more than one point, then the function is 1 -to- 1 .

## Can a linear transformation be not onto and not one-to-one?

In matrix terms, this means that a transformation with matrix A is onto if Ax=b has a solution for any b in the range. If a transformation is onto but not one-to-one, you can think of the domain as having too many vectors to fit into the range.

## How do you know if a transformation is onto?

Let T:Rn↦Rm be a linear transformation. Then T is called onto **if whenever →x2∈Rm there exists →x1∈Rn such that T(→x1)=→x2**. We often call a linear transformation which is one-to-one an injection. Similarly, a linear transformation which is onto is often called a surjection.

## How do you tell if it is a linear transformation?

It is simple enough to identify whether or not a given function f(x) is a linear transformation. Just **look at each term of each component of f(x)**. If each of these terms is a number times one of the components of x, then f is a linear transformation.

## How do you determine if a linear transformation is an isomorphism?

A linear transformation **T :V → W** is called an isomorphism if it is both onto and one-to-one. The vector spaces V and W are said to be isomorphic if there exists an isomorphism T :V → W, and we write V ∼= W when this is the case.

## Is a matrix one-to-one or onto?

**One-to-one is the same as onto for square** matrices

Note that in general, a transformation T is both one-to-one and onto if and only if T ( x )= b has exactly one solution for all b in R m .