Linear Triad Conversion is taking a triad [a 3 tone chord] and playing 3 versions of it on the same set of adjacent strings.
We are converting a chord to a different version of itself, higher on the board [revoicing it].
We can start by playing a Major type melodic arpeggio up a single string [see above]. When we do this, we can see the fret spacing of the chord [any type, but we are looking at Major]. Major chords are built as a 4-3, where the 4 is a Major 3rd [4 half steps = 4 frets] & the 3 is a minor 3rd [3 half steps = 3 frets]. The 3rd is 4 frets away from the root, the 5th is 3 frets away from the 3rd, & the root is 5 half steps away from the 5th [it's a Perfect 4th back to the root]. On the guitar, a half step is one fret.
We can build a triad in a zone by stacking a root, 3rd, and 5th on 3 adjacent strings. When we do this, the chord can have the root on the lowest fretted string, the 3rd on the lowest fretted string, or the 5th on the lowest fretted string. We then use the remaining two upper adjacent strings to fill in the remainder of the chord.
When we look for the next version on the same set of 3 adjacent strings, a linear cycle occurs [3 linears happening at once from different starting points]: the root moves to the 3rd of the chord, the 3rd moves to the 5th, & the 5th moves to the root.
These are sometimes called inversions [Root position = root in bass; 1st inversion = 3rd in bass; 2nd inversion = 5th in bass]. I prefer to call them conversions. We are converting a chord into a form of itself by using some basic numerical spacing. If we converted them using different spacing, this would be converting them to a different chord name/type.
I'm not sure what inversion talk does exactly except to add one more level of naming [which isn't consistent among the set - root position is not an inversion], so again, I prefer the term conversion. Also, when dealing with chords on the higher strings, we often include open strings, so talking about what is 'in the bass' at that point is, I think, taking analyzation one step too uselessly far. What is 'in the bass' is often used for passing chords, such as slash chords, and those situations use a specific nomenclature [slash chords] and talk of which inversion we are playing is often not mentioned, and for good reason. We just don't talk like that. And, once you know how these work, you just know them as the chord name.