Start with 100 equations $\color{#8F00FF}{A}\mathbf{x} = \mathbf{0}$ for $\mathbf{x} = (x_1, …, x_{1oo})$. Suppose elimination reduces the 100th equation to $0 = 0$, so the system is "singular".
(a) Elimination takes linear combinations of the rows. So this singular system has
the singular property: Some linear combination of the 100 rows is $\color{green}?$.(b) Singular systems $\color{purple}{A}\mathbf{x} = \mathbf{0}$ have infinitely many solutions. This means that some linear combination of the 100 columns is $\color{ #FF4F00}{??}$.
(d) For a 100 by 100 singular matrix with no zero entries, describe in words the row picture and the column picture of
$\color{ #9966CC}{A}\mathbf{x} = \mathbf{0}$. Not necessary to draw 100-dimensional space.
My Attempt for (a) and (b): The question postulates that for all $ 1 \leq r \leq 100, \sum_{c = 1}^{100}\color{#8F00FF}{a_{rc}}x_c=0.$ Pictorially:
Hitherto, my definition of "singular" is "dependent rows or columns" (cf P27). So $? = \color{green}{\vec{0}}$ and $?? = \color{ #FF4F00}{\vec{0}}$.
Averred Answer for (d): The row picture has $100$ planes meeting along a common line $\color{#0073CF}{\text{through } \vec{0}}$. The
column picture has $100$ vectors all in the same $\color{#0073CF}{99}$-dimensional hyperplane.
$\Large{1.}$ Why must the common line be $\color{#0073CF}{\text{through } \vec{0}}$? Why not any common line?
$\Large{2.}$ $A$ contains $100$ columns so $100$ column vectors. But $A$ also contains $100$ rows, so how and why $\color{#0073CF}{99}$-dimensional hyperplane?
Please mind that this question is from a germinal section of IoLA, 4th ed by Strang.
Best Answer
elimination takes linear combinations of the rows. So this singular system has singular property some linear combination of the 100 row is non singular singular systems Ax=0 have infinitely many solutions This means that some linear combination of the 100 columns is non zero when Ax=0 then linear combination of rows and columns will be same and donot have value greater than zero,