# Problem Sheet - Linear Algebra MT22, V

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/sheets/5/ · Updated: 2022-11-20 · Tags: uni, problem-sheet

### Flashcards
$$
\left(\begin{matrix} 1 \& 0 \& \frac{1}{3} \& 1 \\\\ 0 \& 1 \& \frac{-2}{3} \& 0 \\\\ 0 \& 0 \& 0 \& 0 \end{matrix}\right)
$$
How would you describe where you put $-1$s for the Minus-1 trick?::
In new rows so that there’s always a $1$ or $-1$ on the diagonal.

$$
\left(\begin{matrix} 1 \& 0 \& \frac{1}{3} \& 1 \\\\ 0 \& 1 \& \frac{-2}{3} \& 0 \\\\ 0 \& 0 \& 0 \& 0 \end{matrix}\right)
$$
How would you rewrite this matrix for the Minus-1 trick?::
$$
\left(\begin{matrix} 1 \& 0 \& \frac{1}{3} \& 1 \\\\ 0 \& 1 \& \frac{-2}{3} \& 0 \\\\ 0 \& 0 \& -1 \& 0 \\\\ 0 \& 0 \& 0 \& -1 \end{matrix}\right)
$$

$$
\left(\begin{matrix} 1 \& 0 \& \frac{1}{3} \& 1 \\\\ 0 \& 1 \& \frac{-2}{3} \& 0 \\\\ 0 \& 0 \& 0 \& 0 \end{matrix}\right) \to \left(\begin{matrix} 1 \& 0 \& \frac{1}{3} \& 1 \\\\ 0 \& 1 \& \frac{-2}{3} \& 0 \\\\ 0 \& 0 \& -1 \& 0 \\\\ 0 \& 0 \& 0 \& -1 \end{matrix}\right)
$$
Here the minus-1 trick has been applied. What are the solutions to the original matrix being $0$?::
$$
\lambda_1 \left(\begin{matrix} \frac{1}{3} \\\\ \frac{-2}{3} \\\\ -1 \\\\ 0 \end{matrix}\right) + \lambda_2 \left(\begin{matrix} 1 \\\\ 0 \\\\ 0 \\\\ -1 \end{matrix}\right)
$$

How can you find the kernel of a linear transformation represented by a matrix $A$?::
Write in reduced row-echelon form and use the minus-1 trick to read off solutions.

How can you find the image of a linear transformation represented by a matrix $A$?::
Write in reduced **column-echelon** form and read off the pivot columns.

What does the linear transformation $(S + T)(v)$ mean where $S, T : V \to W$?::
$$
(S+T)(v) = S(v) + T(v)
$$

What does the linear transformation $(\lambda S)(v)$ mean where $S, T : V \to W$?::
$$
(\lambda S)(v) = \lambda S(v)
$$

What does it mean for a linear transformation $T$ to be idempotent?::
$$
T^2 = T
$$

How, in practice, could you get a matrix in reduced column-echelon form?::
Take the transpose of the matrix, get in RREF, then take the transpose again.

In what situation might you want to use the reduced column-echelon form?::
When determining the image of a matrix.

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