Basis and Dimension

Chapter 5 • Section 5-2

"The building blocks of vector spaces. 🧱"

🕸️ Linear Independence

A set of vectors $\{\vec{v}_1, \dots, \vec{v}_k\}$ is linearly independent if the only solution to:

$$ c_1\vec{v}_1 + c_2\vec{v}_2 + \dots + c_k\vec{v}_k = \vec{0} $$

is the trivial solution $c_1 = c_2 = \dots = c_k = 0$.

If there are non-zero coefficients that make the sum zero, the set is linearly dependent.

📐 Geometric Meaning

In $\mathbb{R}^2$

Two vectors are linearly dependent if and only if they lie on the same line through the origin (i.e., they are parallel).

If independent, they span the entire plane $\mathbb{R}^2$.

In $\mathbb{R}^3$

Three vectors are linearly dependent if and only if they lie on the same plane through the origin (i.e., they are coplanar).

If independent, they span the entire space $\mathbb{R}^3$.

Independence Checker (2D)

Enter two vectors in $\mathbb{R}^2$. If they are multiples of each other (parallel), they are dependent.

Basis and Dimension

Definition: Basis

A set of vectors $\mathcal{B} = \{\vec{v}_1, \dots, \vec{v}_k\}$ is a basis for a subspace $U$ if:

  1. $\mathcal{B}$ is linearly independent.
  2. $\mathcal{B}$ spans $U$ (i.e., $\span(\mathcal{B}) = U$).

Definition: Dimension

The dimension of a subspace $U$, denoted $\dim(U)$, is the number of vectors in any basis of $U$.

Example: The standard basis for $\mathbb{R}^3$ is $\{\vec{e}_1, \vec{e}_2, \vec{e}_3\}$, so $\dim(\mathbb{R}^3) = 3$.

🔍 Finding a Basis

How do we find a basis for the span of a set of vectors? We use the Column Space Algorithm.

Algorithm

  1. Form a matrix $A$ with the given vectors as its columns.
  2. Row reduce $A$ to Row Echelon Form (REF).
  3. Identify the pivot columns in the REF matrix.
  4. The corresponding columns in the original matrix $A$ form a basis for $\text{span}\{\vec{v}_1, \dots, \vec{v}_k\}$.

Example

Find a basis for $U = \text{span}\{(1,2,3), (2,4,6), (1,0,1)\}$.

1. Form matrix $A$:

$$ A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 0 \\ 3 & 6 & 1 \end{bmatrix} $$

2. Row reduce to REF:

$$ \sim \begin{bmatrix} 1 & 2 & 1 \\ 0 & 0 & -2 \\ 0 & 0 & 0 \end{bmatrix} $$

3. Pivots are in columns 1 and 3.

4. Basis is $\{\vec{v}_1, \vec{v}_3\} = \{(1,2,3), (1,0,1)\}$.

🧠 Knowledge Check Win $15

What is the dimension of the subspace of $\mathbb{R}^3$ spanned by $\vec{v}_1 = (1,0,0)$ and $\vec{v}_2 = (0,1,0)$?