Albert and the Four Spaces of a Tiny Matrix

Two small matrices revealing how linear dependence reshapes column space, null space, and orthogonality in real time.

Today Albert explored Gilbert Strang’s four fundamental spaces using two very small matrices.

The first matrix was:

A = [1 2 3]

Even though it is extremely small, it already contains all four fundamental spaces.

Albert first examined the column space of Aᵀ:

C(Aᵀ) = span{ [1, 2, 3]ᵀ }

Then he solved the homogeneous equation:

x + 2y + 3z = 0

and derived the null space:

N(A) = span{ [1, 4, -3]ᵀ, [1, 7, -5]ᵀ }

He also identified the column space of A:

C(A) = span{ [1] }

and observed that the left null space is trivial:

N(Aᵀ) = {0}

meaning there is no nonzero vector perpendicular to the column space in this case.

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Then Albert changed the structure of the matrix:

A = [1 2 3] [1 2 3]

Now the matrix has two identical rows.

This small change dramatically changes the geometry.

The left null space immediately appears:

N(Aᵀ) = span{ [1, -1]ᵀ }

because the two rows cancel each other:

x + y = 0

This was the key moment: linear dependence creates a new direction in the space of solutions.

Albert could directly see how repeating a row changes rank and introduces new null structure.

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Through these two tiny matrices, he explored the four fundamental spaces:

• column space • row space • null space • left null space

not as abstract definitions, but as objects that visibly change when the matrix changes.

This is the power of small examples in linear algebra: they make structure observable.

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Today’s large-scale AI systems, graphics engines, and language models all depend on massive matrix operations.

But the core ideas—rank, dependence, orthogonality, dimension—begin with exactly this kind of simple structure.