Linear Independence and Dimension

Chapter 6 • Section 6-3

"Finding the essential building blocks of any space. 🧱"

🕸️ Linear Independence

A set of vectors $\{\vec{v}_1, \dots, \vec{v}_k\}$ in $V$ is linearly independent if:

$$ c_1\vec{v}_1 + \dots + c_k\vec{v}_k = \vec{0} \implies c_1 = \dots = c_k = 0 $$

If there is a non-trivial solution, the set is dependent.

Example: Polynomials

The set $\{1, x, x^2\}$ is independent in $\calP_2$.

If $a(1) + b(x) + c(x^2) = 0$ (the zero polynomial), then $a=0, b=0, c=0$.

Independence Checker (Polynomials)

Check if two polynomials $P(x)$ and $Q(x)$ are linearly independent.

Polynomial $P(x)$

$x +$

Polynomial $Q(x)$

$x +$

🏛️ Fundamental Theorems

Fundamental Theorem

If $V$ is spanned by $n$ vectors, then any independent set in $V$ has at most $n$ vectors.

Implication: You can't have more independent vectors than spanning vectors.

Invariance Theorem

If $V$ has a basis of $n$ vectors, then every basis of $V$ has exactly $n$ vectors.

This is why "dimension" is a well-defined number!

Unique Representation Theorem

If $B = \{\vec{b}_1, \dots, \vec{b}_n\}$ is a basis for $V$, then every vector $\vec{v} \in V$ can be written as a linear combination of $B$ in exactly one way.

Basis and Dimension

Dimension

The dimension of a vector space $V$, denoted $\dim(V)$, is the number of vectors in any basis of $V$.

$\dim(\calP_n) = n+1$

Basis: $\{1, x, x^2, \dots, x^n\}$. Count is $n+1$.

$\dim(\M_{mn}) = mn$

Basis: Matrices with one 1 and zeros elsewhere. Count is $m \times n$.

🧠 Knowledge Check Win $15

What is the dimension of the space of $2 \times 2$ matrices, $\M_{22}$?