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## Homework Statement

##S## is a linear transformation and ##\{u_{1},u_{2}\}## is a basis for the vector space.

$$

S(u_{1})=u_{1}+u_{2}\\

S(u_{2})=-u_{1}-u_{2}

$$

I would like to find a basis of the null space and range of ##S##.

## Homework Equations

In my text, it says that the proper matrix representation of ##S## is

$$

\left(\begin{array}{cc}

1 & -1 \\

1 & -1 \\

\end{array}\right).

$$

## The Attempt at a Solution

I understand that this way of representing S with a matrix will preserve equality when we compose two linear transformations in matrix from but it is confusing me. I found a supposed basis for the null space as follows. I first put the matrix in echelon form:

$$

\left(\begin{array}{cc}

1 & -1 \\

1 & -1 \\

\end{array}\right) \rightarrow \left(\begin{array}{cc}

1 & -1 \\

0 & 0 \\

\end{array}\right).

$$

Then we are left with the equation ##x_{1}-x_{2}=0## so any vector in the null space has the form ##(\lambda,\lambda)## where ##\lambda## is an arbitrary number. From the definition of the null space we should have that ##S\cdot(\lambda,\lambda)=0## and this holds when I put ##(\lambda,\lambda)## into column form and multiply it by the matrix of ##S## but when I try to plug ##(\lambda,\lambda)## into ##S(u_{1},u_{2})=(u_{1}+u_{2},-u_{1}-u_{2})##, things don't seem to work out anymore. Did I do something wrong when finding the basis or am I thinking about the null space in the wrong way?

I would also like help on finding a basis for the range. I know that it is the same as a basis for the column space but again, do I use the columns in the matrix of ##S## as I have defined it above?

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