# Notes - Linear Algebra I MT22, Chapter 1

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/notes/chapter-1/ · Updated: 2022-12-30 · Tags: linear-algebra, mt22, uni

- [Course - Linear Algebra MT22](https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/)

### Flashcards
How could you prove that applying EROs does not change the solution set of a system of equations?::
Note that the matrices representing all EROs are invertible so if $Ax = b$ then $EAx = Eb$ follows.

What is the necessary and sufficient condition for a system of linear equations represented by a matrix $(A|b)$ in RREF to have no solutions?::
The last non-zero row of $(A|b)$ is $(0\text{ }0\ldots0|1)$.

What is the necessary and sufficient condition for a system of linear equations represented by a matrix $(A|b)$ in RREF to have a unique solution?::
The non-zero rows of $A$ from the identity matrix.

The following is an augmented matrix (imagine there’s a line for the last column):
$$
\left(\begin{matrix} 1 \& 2 \& 0 \& 0 \& 3 \\\\ 0 \& 0 \& 1 \& 0 \& 2 \\\\ 0 \& 0 \& 0 \& 1 \& 1 \end{matrix}\right)
$$
What vector specifes the family of solutions in terms of a parameter $\lambda$?::
$$
\left(\begin{matrix} 3+2\lambda \\\\ -\lambda \\\\ 2 \\\\ 1 \end{matrix}\right)
$$

The following is an augmented matrix (imagine there’s a line for the last column):
$$
\left(\begin{matrix} 1 \& -2 \& 0 \& 2 \& 3 \\\\ 0 \& 0 \& 1 \& 1 \& -2 \\\\ 0 \& 0 \& 0 \& 0 \& 0 \end{matrix}\right)
$$
What vector specifies the family of solutions in terms of $\alpha$ and $\beta$?::
$$
\left(\begin{matrix} 3-2\alpha+2\beta \\\\ -\alpha \\\\ -2+\beta \\\\ -\beta \end{matrix}\right)
$$

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