30: Chapter 30 State Space

Linear Algebra's first page starts with vectors.

Vectors are something Jiang Lin learned in high school.

Plane vectors and space vectors are nothing more than an arrow pointing from point A to point B.

Put into a coordinate system, they are a pair of numbers, or three numbers.

Too familiar.

Yet precisely because he was familiar with them, Jiang Lin's internal alarm bells rang wildly.

Having been beaten black and blue all the way through Advanced Mathematics, he knew the nature of university textbooks all too well.

The more friendly-looking an acquaintance, the more likely they were to bare their fangs at you after turning the page.

That little bit of high school experience with differentiation and calculating areas was only enough to stand outside the door and beg in front of real calculus.

Therefore, when this arrow swaggered onto the first page, Jiang Lin did not rush to claim it as an old friend.

He wrote a few lines on the e-ink handwriting tablet.

[Linear Algebra, Phase One.]

[Don't rush to curse yet.]

[Let's see how it actually wants me to reconstruct the world.]

Outside the window, the cold season of the Wasteland was still lingering on its last legs.

Over the past six years, Advanced Mathematics and General Physics had helped him stand firm in the Wasteland.

But he soon crashed into a new wall.

The Wasteland was never a single-variable test paper.

Whether the Stone House was cold or not could not be judged solely by how many degrees it was outside.

How much electricity the solar panels generated didn't entirely depend on whether the sun cooperated.

The life and death of the farmland was even more of a collective carnival of soil, moisture, acid rain, seeds, and temperature.

Behind every survival system dragged a long string of muddy variables.

And Linear Algebra seemed to be born for this mess.

In the spring of the seventh year, Jiang Lin made his first attempt to drag vectors into the mire of reality.

He created a new document: [State Vector _ Version 1].

First line: [S _ house] (Stone House state).

He casually filled in the data from a certain night in the cold season.

Sleeping area 6.1 ℃, outdoors -4.6 ℃, wind force level 3, door crack status 0, remaining fuel 420 grams, humidity slightly high.

Then, he pulled up a set of data after the thermal insulation modification and lined it up below.

Two sets of data were listed on the screen. Jiang Lin stared at them for a long time, rubbing his hands together in excitement.

He understood: vectors weren't just arrows; vectors were slices of a set of states.

But this wave of excitement lived for less than ten minutes.

He had originally wanted to use the formula in the textbook to calculate the distance between the Stone House states over the past two days.

Subtracting corresponding coordinates, squaring, summing, and taking the square root.

Tapping on the keyboard, he even felt a little smug.

With a tap of the Enter key, an absurdly large value popped up on the screen.

Jiang Lin's brain went buzz.

He had ignored a fatal problem: fuel was calculated in grams, often reaching several hundred.

Temperature was calculated in degrees, differing by a few degrees at most.

Under the crushing weight of those hundreds of grams squared, the changes in temperature and wind force didn't even splash a single ripple.

This meant that as long as he picked up two extra catties of rotten wood on a certain day, even if the door crack was leaking wind, the stove fire was out, and his lips were turning purple from the cold inside the house, this damn mathematical formula would still smilingly judge:

Today's condition is excellent.

"Fine my ass."

Jiang Lin cursed inwardly, leaning back against the stone chair. The heart that had just warmed up was doused with a sizzle by the cold air of the Wasteland.

This was the first heavy blow Linear Algebra dealt him.

It allowed you to write the world as a set of numbers, but the premise was that you first had to strip off their various physical units and compress them into proportions of the same scale.

Otherwise, temperature and fuel mass would forever be like chickens talking to ducks, completely unqualified to sit at the same table in the same geometric space.

Only later did he find out that this step wasn't called casually shrinking numbers.

It was called nondimensionalization.

Temperature, fuel, wind force, and humidity first had to be turned into comparable proportions, and then reweighted according to their impact on survival status.

Otherwise, so-called distance was merely units pretending to be mathematics.

Biting on the end of his pen, Jiang Lin added a sentence in the document.

[You can't just grab a bunch of numbers to line up and call them a vector; undefined variables are just a bunch of schizophrenic nonsense.]

In the late spring of the seventh year, he crashed into matrices.

The multiplication rules were as simple as elementary school arithmetic: multiplying rows and columns and then adding them up. He got over a dozen questions right in a row, but his stomach felt increasingly sour.

Mapping, transformation. The teacher in the video spoke with such casual ease, while Jiang Lin listened in utter confusion. He found that he was once again using "tactical diligence" to cover up "strategic brain crashes." It was just like grinding test papers in high school—getting full scores without knowing what these letters could actually do in reality. Right now, he was like a human abacus, crackling away as he multiplied a bunch of new matrices, but then what? Could it help him grow an extra potato?

That evening, he sat at the doorway of the Stone House letting the wind blow, staring blankly in the direction of the withered riverbed. The fifteen-square-meter Stone House was not a uniform rectangular box; when the stove was lit, the study heated up quickly, and after the fire went out, the storage area cooled down the fastest because it was close to the door crack. They pulled at each other and transferred heat to each other.

Jiang Lin suddenly stood up and rushed back to the stone table.

If the temperatures of the three temperature zones—living area, study, and storage area—were written as a state vector T(t), then in a sufficiently short period of time Δt, could the temperature of the next moment be approximately derived from the temperature of the previous moment?

Jiang Lin did not rush to say yes.

The actual situation must be much dirtier than the example problems in the textbook.

Yet if one only took a very small time step and first compressed those violent changes into the external input term, then at least the first-tier approximation could be written as:

T(t + Δt) = A(t)T(t) + b(t)

Of course, this A wouldn't be fixed forever.

Whether the door crack was blocked, whether the walls were damp, and whether the wind direction had changed would all alter it.

Nor was b a constant dropped from the sky.

Stove fire, outdoor temperature, human body heat dissipation, solar radiation, and even the bit of heat absorbed by the walls during the day could all be crudely crammed into this external input term.

This was crude.

Yet even if it was just a crude first-order approximation, for the first time, Jiang Lin saw how states evolved.

Looking again at A representing the matrix, a tingling sensation crawled up Jiang Lin's back.

A matrix was not an Excel spreadsheet; a matrix was an action.

It was a cruel set of rules dictating how the current state should transform into the appearance of the next moment.

The first row determined the living area, the second row determined the study, and the third row determined the storage area.

The numbers on the diagonal represented how much of its own old temperature each area retained.

The numbers off the diagonal represented the influence of other areas on it.

Multiplying a matrix by a vector was using physical laws to forcefully push the old world toward a new world.

As this breath relaxed, the path ahead suddenly widened.

He no longer rushed to catch up on the progress, but instead crammed everything in the Wasteland into matrices to polish them to a patina.

Not to calculate everything accurately.

Rather, it was to see clearly along which channels vibrations would propagate when a variable was changed.

Wiping the solar panels changed the dust component.

Changes in dust affected power, changes in power affected how long the computer could stay on at night, and changes in computer time directly intervened in study progress.

In the Wasteland, if you moved any single variable, all other variables would tremble along with it.

In human beings' most convenient primitive coordinates, the Real World was never a diagonal matrix.

Yet precisely because of this, Linear Algebra had meaning.

What it wanted to find was not a bigger table.

What it wanted to find was the true set of coordinates hidden behind the coupling.

In the summer of the seventh year, Jiang Lin crashed into basis.

The textbook said that the same vector would have completely different coordinate representations under different bases.

This sentence made Jiang Lin furrow his brows straight into a frown.

The instinct carved into his bones by high school education was to search for the single correct answer.

For the exact same thing, why should the numbers change?

After they changed, which one was actually real?

Carrying this irritability, he walked out of the Stone House and looked at the solar panel trembling slightly in the wind and sand.

The wind was blowing from the northwest.

If standing in map coordinates, this gust of wind could be split into an east-west component and a north-south component.

But if standing in the solar panel's own coordinates, it turned into another set of things.

The wind speed component perpendicular to the panel surface and the wind speed component along the panel surface.

The former determined how much it was frontally pressed by the wind.

The latter determined how the wind and sand scraped past along the panel surface.

Jiang Lin took a deep breath of the dry air of the Wasteland, hurried back to the stone table, and typed a line of text in the document.

[Stop asking what the correct answer on the Wasteland is.]

[University mathematics is forcing me to admit: first ask yourself, what coordinate system are you actually standing in to look at this problem?]

At this moment, Jiang Lin experienced a slight sense of dizziness.

By the autumn of the seventh year, while organizing farmland data, Jiang Lin ushered in his third wave of mental critical hits.

The screen was full of data, from surface moisture content to the number of acid rain occurrences, sprawling across dozens of columns.

At first glance, it looked scientific as hell, but as soon as Jiang Lin examined the logic a little, he felt something was wrong.

He began checking them item by item.

Surface moisture content was related to the moisture content at a depth of ten centimeters, but they weren't completely identical.

The average temperature of the past seven days was related to effective sunlight, but they also pulled against each other.

Compost maturity was related to the years of plot ripening, but some old plots might not be better than newer plots carefully treated later on due to crude early operations.

There were no clean relationships between the variables; they were like a group of suspects locked in a spreadsheet colluding with each other.

Every column was screaming that it was important, and every column was secretly colluding with other columns under the table.

The textbook called this linear correlation.

Immediately following popped out a chilling word: rank.

The video teacher said that in the crudest sense, rank could be understood as the effective independent amount of information.

Jiang Lin straightened up all at once.

Because these few words were like a slap in the face, shattering the illusion he had felt over the past few years while feeling smug in front of a pile of heavy data.

Just because the farmland state table spanned two screens didn't mean its rank was very high.

Writing a hundred columns of variables sprawlingly was very likely just a bunch of moisture and temperature wearing a dozen different vests and jumping back and forth repeatedly.

Of course, there was even more disgusting stuff in real data.

Noise.

A tiny measurement error was enough to turn a spreadsheet that should have been low-rank into full rank computationally.

Therefore, what he should truly look at was not just the chilling rank, but which directions contributed the main changes and which directions were merely noise shaking.

Having many variables wasn't scary.

What was scary was that he had no idea which variables were providing information and which variables were merely creating illusions.

His past life had also been like this.

Test papers, notes, and error books were overwhelming; they looked broad, but in reality, they were all spinning in place on the exact same few knowledge points.

The amount of effective information was pathetically low.

Jiang Lin took out his paper time capsule.

[Linear Algebra: Rank.]

[Having many variables does not equal having much information.]

[Having many notes does not equal having much understanding.]

[What truly matters are independent directions and the amount of effective information.]

When the first cold wind of winter blew past, Jiang Lin began studying systems of linear equations and determinants.

Solving equations in high school was a matter of course, but college Linear Algebra sneered and tore up this guarantee.

Reality was often unsolvable.

He tried to calculate the main cause of the Stone House's cooling and the key to increasing potato yield, only to find the data frantically beating each other up. Today's temperature didn't match yesterday's records, and wet weight and dry weight were mixed together.

It wasn't that he couldn't calculate it; the data itself was rotten to the core.

Reality was a sloppy operation leaking wind from all sides, and it absolutely refused to take the blame for your crude records.

Only later did he know that most systems of equations in the Real World didn't give precise solutions to begin with.

When you asked it which exact answer it was, it would only fling a bunch of mutually fighting observation values at you.

What should truly be asked was not whether there was a perfect answer.

Rather, among all imperfect answers, which one minimized the overall error.

That thing was called residual.

That path led to least squares.

With reddened eyes, Jiang Lin did not delete all the records.

He first dragged the data with obviously confused calibers, unknown units, and missing dates and condition descriptions into the [Discarded Data] folder.

As for the rest that could still be saved, he created a separate folder.

[Residual too large, pending re-evaluation].

Useless was useless; self-moved sentimentality couldn't be eaten as meals.

And the determinant gave him an ultimatum.

In linear transformation, the determinant was the scaling factor of volume.

If det = 0, it meant the space was directly flattened, dimensions collapsed, and information was lost.

This was irreversibility at the information level.

Three dimensions compressed into two dimensions, two dimensions compressed into a line.

You only got the flattened shadow and could never restore the original volume.

Jiang Lin suddenly thought of how there were many similar moments in the Wasteland.

Seeds molding, batteries deeply discharging, water sources contaminated, bodies crossing the line of death.

They weren't truly having a determinant of zero.

But they were all reminding him that once certain operations lost critical states, they could never be retreated from.

From then on, Jiang Lin learned to distinguish between controllable trial-and-error and irreversible trial-and-error.

Don't bet your life on operations that would flatten all retreats.

In the summer of the eighth year, he chewed on the hardest bone: eigenvalues and eigenvectors.

Direction unchanged, only stretching?

Looking at this tongue twister, Jiang Lin only felt his brain turning numb.

He still returned to the Stone House at the very beginning.

Three temperature zones, three temperatures—they could still be written as a state vector.

Temporarily removing external inputs like the stove fire and door cracks, he looked only at the internal heat transfer among the three temperature zones.

If the temperatures of the living area, study, and storage area were completely identical, then in the next moment, they would still most likely rise together and fall together.

This was one direction.

If the study was hotter than the living area and the storage area was colder than them, heat would flow along the walls and door cracks, and the temperature difference would be smoothed out bit by bit.

This was yet another direction.

So eigenvectors weren't metaphysics.

They were the most natural directions of change for the system.

Eigenvalues told him whether the state in this direction was retained, attenuated, or amplified.

Later, on a certain day when a gale raged, the wooden support of the solar panels let out a squeaking, miserable groan in the wind.

Jiang Lin thought of another even more dangerous word.

Mode.

The support didn't shake the exact same way no matter how it was shaken.

Some shaking shapes were quickly swallowed up by the structure and damping.

Some shaking shapes, however, seemed to step right on the bone joints of the system; as long as the rhythm of the external wind force approached its natural frequency, energy would be poured in stroke by stroke.

That wasn't ordinary shaking.

That was resonance.

And behind the modes stood eigenvalues and eigenvectors.

Thinking of this, when Jiang Lin looked at the Wasteland again, he found that many things seemed to have developed tempers of their own.

If all efforts were smashed onto the wrong characteristic directions—such as rote memorization and staying up late in fake diligence—not only would the system not get better, but it would accelerate its collapse.

In the early winter of the eighth year, Jiang Lin made a comprehensive compilation.

He saved those crude models full of errors and named them [State Space _ 8th Year Compiled Edition].

There was no tutor to grade it, nor any internet to show it off, but this crude system was hammered out brick by brick by him alone in the darkness.

Night descended, and the embers in the stove fire lingered on their last legs.

Jiang Lin opened the paper time capsule, his wrist slightly aching from holding the pen for a long time.

[Linear Algebra, first round ends.]

[Not proficient, but enough to use.]

[Advanced Mathematics taught me to see changes, Physics taught me to see laws, and Linear Algebra taught me to see structures.]

[The Wasteland is never a smooth curve, but a state space riddled with traps.]

The ink dried completely.

Jiang Lin closed the file bag, rubbed his icy knuckles, and the mouse cursor hovered over the next unnamed folder.

Lying in the folder was an even more unreasonable name.

If Linear Algebra taught him to see structures...

Then the next thing was going to force him to admit another matter.

Every single number given by the Real World might be lying.