Chapter 7 — Linear Independence, Bases, and Dimension
Span tells us what can be built. Linear independence tells us whether our building blocks are redundant. A basis is a minimal, nonredundant spanning set, and dimension counts how many directions are truly needed.
Dependence means redundancy
Linearly dependent
There is a nontrivial relation
where not all $c_i$ are zero.
Linearly independent
The only solution to
is $c_1=\cdots=c_k=0$.
Independence checker
Place the vectors as columns of a matrix. The vectors are independent exactly when every column is a pivot column.
Interactive: test independence by row reduction
Basis and dimension
Minimal
No vector in a basis is redundant.
Complete
A basis spans the whole space or subspace.
Coordinate system
Every vector has a unique coordinate vector in a basis.
Find pivot columns
Interactive: basis for a column space
Coordinates relative to a basis
If $B=[v_1\ \cdots\ v_n]$ is a basis matrix, then the coordinate vector $c$ of $x$ satisfies $Bc=x$.
Interactive: compute coordinates in a basis
Geometry in $\mathbb R^2$
Two vectors in $\mathbb R^2$ form a basis exactly when they are not parallel.
Interactive: visualize two candidate basis vectors
Chapter summary
- Dependence means there is a nontrivial relation $c_1v_1+\cdots+c_kv_k=0$.
- Independence means the only such relation is the trivial one.
- A basis is an independent spanning set.
- The pivot columns of the original matrix form a basis for the column space.
- Dimension is the number of vectors in a basis.
Quick checks
Can three vectors in $\mathbb R^2$ be independent?
No. Too many vectors guarantee dependence.
Can a set containing the zero vector be independent?
No. The zero vector gives a nontrivial dependence relation.
Why use original pivot columns, not rref columns, for a column-space basis?
Row operations change the columns and their column space. Pivot positions are found in rref, but the basis vectors come from the original matrix.