Subspaces — four fundamental subspaces of a matrix: Visual walkthrough

Step 0 — What is a "vector" and what is " $A x$ "? (nothing assumed)

Our matrix $A$ takes an input arrow $x = [x_{1} x_{2}]$ and produces an output arrow. The rule is combine the columns:

the machine A input arrow x = x_{1} column a_{1} [12] + x_{2} column a_{2} [24] .

$x_{1}$ = how much of the first column arrow you take.
$x_{2}$ = how much of the second column arrow you take.
The output is the tip of the two scaled arrows placed head-to-tail.

WHY read $A x$ this way? Because it turns an abstract "matrix times vector" into a picture of mixing two fixed arrows. That single idea generates every subspace below.

Figure — Subspaces — four fundamental subspaces of a matrix

Step 1 — Draw the two columns and look at them

WHAT we do: plot $a_{1} = [12]$ and $a_{2} = [24]$ .

WHY: the output of the machine is always a mix of these two, so the set of reachable outputs is decided entirely by what these two arrows point at.

PICTURE: notice $a_{2} = 2 a_{1}$ — column 2 is column 1 stretched by 2. They lie on the same line. So mixing them never escapes that line. The blue line is the whole column space: a 1-dimensional line through the origin, not the full plane.

Step 2 — Which outputs are FORBIDDEN? (the red zone)

WHAT: ask "can the output ever leave the blue line?" No. Any $b$ off the blue line can never be written as $A x$ .

WHY it matters: this is exactly the statement "$Ax=b$ is solvable $⟺ b \in C (A)$ ". A target off the line has no recipe.

PICTURE: the plane splits into two coloured regions — the blue line of reachable outputs, and everything else (drawn red) that is unreachable. The output space $R^{2}$ is carved in two.

Step 3 — Which inputs get CRUSHED to zero? (the null space)

WHAT we do: solve $A x = 0$ . Since both rows are $[12]$ (row 2 $= 2 \times$ row 1), we need only:

coeff of x_{1} 1 x_{1} + coeff of x_{2} 2 x_{2} = 0 ⟹ x_{1} = - 2 x_{2} .

Pick $x_{2} = 1$ : the special solution is $[- 2 1]$ .

WHY: one equation in two unknowns leaves one free variable, so $dim N (A) = n - r = 2 - 1 = 1$ — a line of crushed inputs. See Rank–Nullity Theorem.

PICTURE (in the input space): the green line through $[- 2 1]$ . Every arrow on it, fed into the machine, comes out as the zero arrow.

Quick check of the crush: $A [- 2 1] = [1 (- 2) + 2 (1) 2 (- 2) + 4 (1)] = [00] . ✓$

Step 4 — The rows, and why they are perpendicular to the crush

WHAT we do: draw the row arrow $r_{1} = [12]$ (orange) in the same input plane as the green null line.

WHY the tool "dot product": $A x = 0$ means each row dotted with $x$ is zero. The dot product $r \cdot x = r_{1} x_{1} + r_{2} x_{2}$ answers exactly one question — "are these two arrows perpendicular?" — because it equals zero precisely when they meet at $9 0^{\circ}$ . That is why dotting, not any other operation, exposes the geometry here.

row dotted with input r_{1} \cdot x = = - 2 + 2 (1) (- 2) + (2) (1) = 0 \Rightarrow orange ⊥ green.

PICTURE: orange row line and green null line cross at a perfect right angle. This is $N (A) ⊥ C (A^{T})$ made visible. See Orthogonal Complements.

Step 5 — The fourth subspace: forbidden outputs, found by transposing

WHAT we do: solve $A^{T} y = 0$ . Columns of $A$ are the rows of $A^{T}$ ; both are multiples of $[12]$ , giving one equation:

from a_{1} 1 y_{1} + from a_{2} 2 y_{2} = 0 \Rightarrow y = [- 2 1] .

WHY: this direction is perpendicular to every column, so it is perpendicular to the whole blue column line. It names the red forbidden zone: outputs pile up along blue, and the direction you can never reach is exactly this red arrow.

PICTURE (in the output space): blue column line and red left-null line cross at $9 0^{\circ}$ . $N (A^{T}) ⊥ C (A)$ , and $dim N (A^{T}) = m - r = 2 - 1 = 1$ .

Orthogonality check (column $⊥$ left-null): $[12] \cdot [- 2 1] = - 2 + 2 = 0. ✓$

Step 6 — Edge case: the full-rank machine (nothing is crushed)

WHAT: swap to $B = [2113]$ , whose columns point in different directions.

WHY show this: to prove the pictures above were a special (rank-deficient) case. When columns are independent, the story changes shape and you must not carry over old intuitions.

PICTURE: the two blue columns now span the whole plane. Nothing is forbidden ( $C (B) = R^{2}$ , red zone vanishes). The only input crushed to zero is the origin itself, so the green null "line" shrinks to a single point.

r = 2 = m = n \Rightarrow dim N (B) = n - r = 0, dim N (B^{T}) = m - r = 0.

The one-picture summary

Two planes side by side. Left = input space $R^{2}$ : orange row line and green null line at $9 0^{\circ}$ , filling it. Right = output space $R^{2}$ : blue column line and red left-null line at $9 0^{\circ}$ , filling it. The machine $A$ arrows across the middle, sending green $\to$ origin (crush) and everything else $\to$ blue line.

Recall Feynman: retell the whole walkthrough

I drew the machine's two columns and noticed they lay on one line — so the outputs it can make (blue) form just a line, and everything off that line (red) is forbidden juice. Then I asked which inputs the machine flattens: solving $A x = 0$ gave a green line of "wasted recipes". I drew the machine's rows (orange) in the same input plane and dotted a row with a null vector — it came out zero, meaning orange and green cross at a right angle and together fill the input plane. Transposing gave the last piece: a red direction perpendicular to the blue outputs, naming exactly the forbidden zone. Finally I swapped in a matrix whose columns spread apart — now nothing is forbidden and nothing is crushed, showing the earlier squeeze was the special rank-1 case. Two lines in, two lines out, each pair at perfect right angles.

Recall

What does reading

A x

x_{1} a_{1} + x_{2} a_{2}

let you see?

The output is a mixture of the fixed column arrows, so reachable outputs = span of columns =

C (A)

In our matrix

A = [1224]

, why is

C (A)

only a line?

Column 2

= 2 \times

column 1, so the two columns share one direction; rank

= 1

Why does dotting a row with

x

expose the geometry of

N (A)

(A x)_{i} = r_{i} \cdot x

, and a dot product is zero exactly when the arrows are perpendicular — so

A x = 0

means

x ⊥

every row.

What happens to the null space when

A

is full rank (

r = n

It collapses to the single point

{0}

; nothing is crushed.

Which subspace names the "forbidden output" direction in

R^{m}

The left null space

N (A^{T})

, perpendicular to the column space.

Connections

Rank of a Matrix — the one number $r$ that fixed every dimension in these pictures.
Rank–Nullity Theorem — the green line's dimension $n - r$ came straight from here.
Orthogonal Complements — the right angles in Steps 4 and 5.
Solving Ax=b — Step 2's red forbidden zone is exactly the unsolvable targets.
RREF and Pivots — how we found the pivot/free split behind the drawings.
Least Squares & Projections — what to do when your target lands in the red zone.