Foundations — Subspaces — four fundamental subspaces of a matrix

This page assumes nothing. We build every symbol the parent note leans on, in an order where each new idea only uses ideas already defined.

1. ${\mathbb{R}}^n$ — the space where vectors live

Plain words: the little $\mathbb{R}$ means "real numbers" (like $-2$, $0.5$, $\pi$). The superscript $n$ counts how many slots each vector has.

The picture: ${\mathbb{R}}^2$ is the flat page you draw on — every point $(x,y)$ is an arrow from the origin $(0,0)$ to that point. ${\mathbb{R}}^3$ is the room you sit in. Higher $n$ we cannot draw, but the rules stay identical.

Why the topic needs it: the parent says two subspaces "live in ${\mathbb{R}}^n$" (inputs) and two "live in ${\mathbb{R}}^m$" (outputs). That sentence is meaningless until you know ${\mathbb{R}}^n$ is just "the space of $n$-slot vectors."

2. A vector, and the origin $0$

The picture: the origin is the single anchor point $(0,0)$ where every arrow begins. The bold $0$ is that anchor itself, viewed as a (degenerate) arrow.

Why the topic needs it: the very first rule of a subspace is "contains $0$." A subspace is a "flat thing through the origin." Without $0$ you cannot even state rule 1.

3. Two vector operations: add and scale

The picture: addition is "tip-to-tail" — walk along the first arrow, then along the second; where you end is the sum. Scaling by $c$ stretches the arrow (if $|c|>1$), shrinks it (if $|c|<1$), or flips it around the origin (if $c<0$).

Why the topic needs it: subspace rules 2 and 3 are exactly these two operations. "Closed under addition/scaling" means: do these operations and you never leave the set. Every subspace is built only from adding and scaling.

4. Linear combination and $\text{span}$

The picture: the span of one nonzero vector is the line through it. The span of two non-parallel vectors is the whole plane they sweep out. Span is "everywhere you can reach by mixing."

Why the topic needs it: the column space is defined as $\text{span}$ of the columns, and the row space as $\text{span}$ of the rows. Span is the word that turns "a few vectors" into "a whole subspace."

5. The dot product $u\cdot v$ and "perpendicular"

Why this tool and not another? The topic's deepest claim is that the subspaces come in perpendicular pairs. To say "perpendicular" with numbers you need one gadget that returns $0$ precisely at a right angle — the dot product is that gadget. No angle-measuring, no protractor: a single sum tells you if two arrows meet at $90^{\circ }$.

The picture: slide one arrow so both start at the origin. If they make a right angle, their dot product is $0$. If they lean the same way it is positive; opposite ways, negative.

Why the topic needs it: the whole of "Section 4: Orthogonality" is written in dot-product language. "$x$ is orthogonal to every row" means every row dotted with $x$ gives $0$.

6. The matrix $A$ and the product $Ax$

Plain words: feed in the list $x$; each entry of $x$ says "how much of that column to use." Add the scaled columns and you get the output.

The picture: think of $A$ as a machine. In goes an $n$-slot vector, out comes an $m$-slot vector. The columns of $A$ are the machine's "building blocks"; $x$ is the recipe of how much of each block to stir in.

Why the topic needs it: everything is $Ax$.

• Column space = all outputs $Ax$ (the reachable juices).
• Null space = inputs with $Ax=0$ (recipes that make nothing). Once you accept "$Ax$ = mix of columns," the definitions read themselves.

7. The transpose $A^T$

The picture: tip the matrix onto its side. Row 1 becomes column 1.

Why the topic needs it: the row space is defined as $C(A^T)$ — "the column space of the flipped matrix" — because the rows of $A$ are the columns of $A^T$. Likewise the left null space is $N(A^T)$. Transpose is the trick that lets us reuse "column space" and "null space" language for rows.

8. Rank $r$, pivots, and dimension

The picture: rank counts how many arrows actually point in new directions. If column 2 is just twice column 1, it adds no new direction — it does not raise the rank.

Why the topic needs it: the single number $r$ controls all four dimensions: $\dim C(A)=r,\dim C(A^T)=r,\dim N(A)=n-r,\dim N(A^T)=m-r.$ See Rank of a Matrix and Rank–Nullity Theorem for the machinery; here we just note that without "rank" the parent's Section 3 has no vocabulary.

9. Direct sum $\oplus$ and orthogonal complement

The picture: in ${\mathbb{R}}^3$, a plane through the origin and the line through the origin perpendicular to it are orthogonal complements — no shared arrow except $0$, and any vector = (its shadow on the plane) + (its part along the line).

Why the topic needs it: Section 4's punchline is $N(A)\oplus C(A^T)={\mathbb{R}}^n$ with $N(A)\perp C(A^T)$. That single symbol $\oplus$ packages "they fit together with no overlap and no gap."

Prerequisite map

Equipment checklist

Test yourself — each line is prompt ::: answer.

Connections

• Subspaces — four fundamental subspaces of a matrix — the parent this page prepares you for.
• Rank of a Matrix — the number $r$ built in Section 8.
• Rank–Nullity Theorem — where $\dim N(A)=n-r$ comes from.
• Orthogonal Complements — deepens Section 9.
• RREF and Pivots — how pivots (Section 8) are found.
• Solving Ax=b — uses $Ax$ from Section 6.
• Least Squares & Projections — uses the dot product and $\oplus$.