Linear Algebra I MT22, Chapter 1

Created: December 30, 2022 | Read markdown | About these notes | View in context | Study these flashcards |

Flashcards

How could you prove that applying EROs does not change the solution set of a system of equations?

eros-preserve-solution-set

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?

rref-no-solution-condition

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?

rref-unique-solution-condition

The non-zero rows of $A$ form 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 specifies the family of solutions in terms of a parameter $\lambda$?

rref-solution-family-one-parameter-example
\[\left(\begin{matrix} 3+2\lambda \\ -\lambda \\ 2 \\ 1 \end{matrix}\right)\]

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 $\beta$?

rref-solution-family-two-parameter-example
\[\left(\begin{matrix} 3-2\alpha+2\beta \\ -\alpha \\ -2+\beta \\ -\beta \end{matrix}\right)\]


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