Intuition The one core ideaA matrix is a machine that eats input vectors and spits out output vectors , and the whole topic is about two questions: what outputs can it reach? (column space) and what inputs does it destroy to zero? (null space).
Everything else — vectors, spans, dimensions, RREF — is just the vocabulary you need to say those two sentences precisely.
Before you can read the parent note, you need to own every symbol it uses. This page builds each one from nothing, in the order they depend on each other. Nothing here assumes you have seen linear algebra before.
A vector is just an ordered list of numbers, written as a column:
x = x 1 x 2 x 3 . x 1 , x 2 , x 3 components (the individual numbers). The subscript is just a name-tag telling you which slot the number sits in.
A vector with 2 components is an arrow in the flat plane , starting at the origin (the point ( 0 , 0 ) ( x 1 , x 2 ) is the arrow's destination.
We need vectors because the matrix machine takes a vector in and gives a vector out . No vectors, no machine.
Two things you can do to vectors — and these two operations are the entire game:
Definition The two legal moves
Scaling (stretching): multiply every component by a number α α x = ( α x 1 , α x 2 , … ) Adding : line up two arrows tip-to-tail; the sum is the arrow to the final tip: ( x 1 + y 1 , x 2 + y 2 , … )
Why only these two? Because "linear" literally means "built only from scaling and adding." Every idea in this topic is one of these two moves wearing a costume.
R n R all real numbers — every point on the number line. The superscript n "lists of length n .
So R 2 R 3 R n n
A matrix is a rectangular grid of numbers. We write A ∈ R m × n A m rows (horizontal lines) and n columns (vertical lines).
A = [ 1 3 2 0 4 5 ] ( m = 2 rows , n = 3 columns ) .
Intuition Read the shape as "output-by-input"The columns (n m row of column-vectors standing side by side :
A = [ a 1 a 2 ⋯ a n ] . the picture the whole topic runs on — remember it.
Why do we care that each column is a vector in R m combinations of those columns . The columns are the machine's whole "stock of ingredients."
Definition Matrix times vector, the honest wayWhen you multiply matrix A x weighted mix of A , where the weights are the components of x A x = x 1 a 1 + x 2 a 2 + ⋯ + x n a n .
Intuition What this looks likeThink of x x 1 x 2 Turning dials and adding = scaling and adding = linear. That's the whole machine.
Notice A x is an output vector, and it is always some combination of the columns. That single fact is the seed of the column space.
Definition Linear combinationA linear combination of vectors a 1 , … , a n c 1 a 1 + ⋯ + c n a n c i exactly what A x
The span of a set of vectors is every linear combination you can make from them — the entire reachable set.
Span of one nonzero arrow = the whole line through it.
Span of two arrows pointing different ways = the whole plane they sweep out.
Span of two arrows on the same line = still just that line (the second added nothing new).
Intuition Why span is the right word for column spaceThe column space is defined as { A x : x ∈ R n } all possible outputs . Since every output is a linear combination of the columns, "all outputs" = "all combinations of the columns" = the span of the columns . That is literally what the column space is.
The word zero vector 0 0 x = 0
Definition Linearly independentA set of vectors is linearly independent if no vector is a combination of the others — nobody is redundant. Equivalently, the only way to combine them to reach 0 can be built from the others, the set is dependent (someone is dead weight).
The dimension of a space is the number of vectors in any basis — how many independent directions it truly has. A line has dimension 1, a plane dimension 2, and so on.
Intuition Why the topic lives or dies on thisThe parent's two headline results — dim Col ( A ) = r dim Null ( A ) = n − r dimension counts . Without "dimension = size of a basis," those boxed formulas are just symbols. This is the ruler the whole topic measures with.
Definition Row reduction & RREFRow reduction is a fixed recipe of legal row moves (swap rows, scale a row, add a multiple of one row to another) that simplifies a matrix without changing the solutions of A x = 0 RREF (reduced row-echelon form): every leading entry is a 1 Row reduction & RREF .
Definition Pivot, free column, rank
A pivot is a leading 1 pivot column holds one.
A free column has no pivot — it corresponds to a variable you can set freely.
The rank r r = dim Col ( A ) Rank of a matrix .
Intuition Why RREF is the tool of choiceWe need a mechanical way to answer "which columns are redundant?" and "which inputs collapse to zero?" without eyeballing. RREF answers both at once: pivot columns mark the independent directions (for Col ), and free columns mark the degrees of freedom for collapsing (for the null space). It is the single engine behind both bases in the parent note.
Common mistake "Free columns are the zero rows."Why it feels right: zero rows are the obvious 'leftover' bits. Why it's wrong: free-ness is a property of columns (inputs), zero rows are about rows (dependent equations). Nullity counts free columns = n − r
Vectors: lists and arrows
R to the n: input and output rooms
Matrix A as columns side by side
Product Ax as a mix of columns
Span equals reachable outputs
Null space and Column space
Everything upstream must be solid before the parent topic makes sense. Downstream from the topic sit Rank–Nullity Theorem , Injective and surjective linear maps , Solving Ax=b , and the Four fundamental subspaces .
Cover the right side and test yourself. If any answer is fuzzy, re-read that section above.
What is a vector, in plain words? An ordered list of numbers, pictured as an arrow from the origin to that point.
What do R n n All lists of n n
For A ∈ R m × n m n
Why can a matrix be read as columns standing side by side? Because A x R m
What is A x A weighted sum of A x x 1 a 1 + ⋯ + x n a n
What is a linear combination? Any sum c 1 a 1 + ⋯ + c n a n
What is the span of some vectors? The set of ALL their linear combinations — the reachable line/plane/space.
When are vectors linearly independent? When none is a combination of the others; only all-zero weights reach 0
What is a basis? A minimal set that both spans the space and is independent.
What is dimension? The number of vectors in any basis of the space.
What is a pivot, and what is the rank? A leading 1 in RREF; rank r
What does RREF preserve and change? Preserves the solution set of A x = 0
Linear independence and basis — the ideas of §5–§6 in full.Rank of a matrix — rank is the pivot count of §7.Row reduction & RREF — the engine of §7.Rank–Nullity Theorem — the counting identity the topic builds toward.Injective and surjective linear maps — trivial null space ⇔ injective.Solving Ax=b — solvable iff b Four fundamental subspaces — where Col and Null sit among all four. 