Notes - Linear Algebra I MT22, Chapter 1

Created: December 30, 2022

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 $(A \vert b)$ in RREF to have no solutions?

The last non-zero row of $(A \vert b)$ is $(0\text{ }0\ldots0 \vert 1)$.

What is the necessary and sufficient condition for a system of linear equations represented by a matrix $(A \vert b)$ 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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