Intuition The one core ideaA collection of arrows is linearly independent when each arrow points in a genuinely new direction that none of the others could reach by stretching and adding. Everything in this topic — vectors, scalars, the equation A c = 0 "is any arrow a leftover copy of the others?"
This page builds every piece of notation the parent note uses, from absolute zero. Read it top to bottom: each item is used by the next.
A vector is an arrow with a length and a direction , drawn from a starting point (the origin). We also write it as a list of numbers — its coordinates — telling us how far to go along each axis.
The two views are the same object. The list ( 3 , 2 ) 3 x 2 y
list of numbers ?
Drawing arrows is fine for a picture, but to compute we need numbers. The list is the arrow written in a language a calculator understands. In this topic we stack these lists side by side to build a table (a matrix), so the number-list view is essential.
We write vectors in bold : v 1 , v 2 , … name tag (vector number 1, number 2), not a power or a coordinate.
v 2 v Why it feels right: subscripts look like exponents.
The fix: v 2 label — the second vector in our list. It has nothing to do with multiplying v
A scalar is an ordinary single number (like 2 − 2 1 0 scale : multiplying a vector by a scalar stretches or shrinks it, and a negative scalar flips it to point the opposite way.
The parent note uses letters c 1 , c 2 , … , c n c i amount of vector v i
Intuition Why we need scalars for this topic"Can vector A be built from the others?" means "can I scale the others and add them up to land on A?" Scaling is half the recipe. The scalars c i
Watch the special values:
c = 1 c = 0 none of it.c = − 1 c = 2
Definition Vector addition (tip-to-tail)To add two vectors, place the tail of the second at the tip of the first. The arrow from the very start to the very end is the sum . In coordinates you just add matching entries: ( a , b ) + ( c , d ) = ( a + c , b + d )
Now combine scaling and adding:
Definition Linear combinationA linear combination of v 1 , … , v n c 1 v 1 + c 2 v 2 + ⋯ + c n v n . linear means we only ever scale and add — no bending, no multiplying vectors together.
Intuition Why this is the heart of everythingThe entire definition of independence is a sentence about linear combinations : "the only linear combination that lands on the origin is the boring one where every dial is zero." If you understand "scale each arrow, add them up," you already understand the machinery. This connects to Span and spanning sets — the span is the set of all linear combinations you can reach.
The zero vector 0 length zero — it starts and ends at the origin, pointing nowhere. Its coordinate list is all zeros: ( 0 , 0 ) ( 0 , 0 , 0 )
0 "which combinations bring me back to where I started?" Landing back on the origin means the arrows perfectly cancel. If the only way to cancel is to take none of anything (c 1 = ⋯ = c n = 0 other way to cancel exposes a redundant arrow.
0 0 Why it feels right: they look almost identical.
The fix: 0 arrow of zero length — a point. Plain 0 scalar . In c 1 v 1 + ⋯ = 0
R n R n all number-lists of length n R 2 2 R 3 3 n dimension , the number of independent directions available.
Intuition Why dimension matters for the topicThere are only n R n n R 3 Basis and dimension for the full story.
A matrix is a rectangular grid of numbers. In this topic we build one by standing our vectors up as columns , side by side:
A = ∣ v 1 ∣ ∣ v 2 ∣ ⋯ ∣ v n ∣ . ∣
c c c = ( c 1 , c 2 , … , c n ) how much of each column we take.
Intuition Why rewrite it asA c = 0 c 1 v 1 + ⋯ + c n v n = 0 system of equations in disguise. Packing it as A c = 0 one powerful machine — row reduction — to solve it, instead of juggling equations by hand. This is a homogeneous system (the right side is 0 null space .
Definition Trivial and nontrivialc = 0 always solves A c = 0 trivial solution .
A nontrivial solution is any solution where at least one c i = 0
Situation
What it means
Verdict
Only c = 0
No arrow is redundant
Independent
Some c = 0
An arrow is a leftover combo
Dependent
Mnemonic Zero only the boring wayIndependent = you can only reach 0 boring way (all dials off). Any exciting way (a dial left on) means dependent.
After tidying a matrix by row reduction (adding/subtracting rows to make a staircase of leading entries called pivots ), the rank is the number of pivots — the count of genuinely independent columns. See Rank of a matrix .
A column without a pivot corresponds to a free variable — a dial you may set to anything , which instantly produces a nontrivial solution.
Intuition Why counting pivots answers the whole question
Pivot in every column (rank = n independent .A pivot missing (rank < n dependent .
Counting pivots is the mechanical, guaranteed test — no eyeballing required.
The determinant det A only when A n R n signed area/volume the columns enclose. See Determinant .
Intuition Why area answers independenceIf two arrows in R 2 zero area — flat, squashed. So:
det A = 0 det A = 0 A invertible .
Common mistake Reaching fordet A The fix: det 3 × 4
Vector = arrow = number list
Scalar = stretch or flip number
Vector addition tip to tail
Linear combination scale and add
Vectors as columns matrix A
A times c equals combination
Homogeneous system Ac = 0
Trivial vs nontrivial solution
Test yourself — reveal only after answering aloud.
I can draw the vector ( 3 , 2 ) Go 3 right along x y ( 3 , 2 )
I know what multiplying a vector by − 2 Doubles its length and flips it to point the opposite way.
I can write out the linear combination c 1 v 1 + c 2 v 2 Take c 1 v 1 c 2 v 2
I know the difference between 0 0 0 0
I can explain why A c A Each entry c i v i A c
I know what "trivial solution" means for A c = 0 The all-zero dial setting c = 0
I know when I'm allowed to use the determinant. Only for square matrices — n R n
I can state why a free variable forces dependence. A free dial can be set nonzero, giving a nontrivial solution to A c = 0
Span and spanning sets — the set of all linear combinations from Section 3.Basis and dimension — dimension of R n Rank of a matrix — pivots counted in Section 8.Determinant — the area/volume test of Section 9.Homogeneous systems and null space — the equation A c = 0 Invertible matrix theorem — independent columns ⇔ invertible ⇔ det = 0 Linear independence — formal definition, testing — the parent this page equips you for. 
