22. Linearly Independent Sets - Bases

Dated: 28-04-2025

Useful Results

• A set¹ containing the zero vector² is linearly dependent.
• A set¹ of two vectors² is linearly dependent if and only if one is a multiple of the other.
• A set¹ containing at least one non zero vector² i.e. \(\{\vec V\}\) is linearly independent when \(\vec V \neq \vec 0\).
• A set¹ of two vectors² is linearly independent if and only if neither of the vectors² is a multiple of the other.

Definition

Let \(\mathbb H\) be a subspace³ of a vector space³ \(\mathbb V\) then an indexed set¹ of vectors³ \(\mathbb B = \{\vec {b_1}, \vec{b_2}, \ldots, \vec{b_n}\} \in \mathbb V\) is a basis for \(\mathbb H\) if

• \(\mathbb B\) is linearly independent set.
• \(\mathbb H = Span\{\vec{b_1}, \vec{b_2}, \ldots, \vec{b_n}\}\)

Example

\[e_1 = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \quad e_2 = \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \quad \dots \quad e_n = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}\]

The set¹ \(\{\vec{e_1}, \vec{e_2}, \ldots, \vec{e_n}\}\) is called the standard basis for \(\mathbb R^n\).

The Spanning Set Theorem

Let \(\mathbb S = \{\vec{v_1}, \vec{v_2}, \ldots, \vec{v_n}\} \in \mathbb V\) be a set¹ such that \(\mathbb H = Span (\mathbb S)\) and if \(\mathbb H \neq \{\vec 0\}\) then some subset¹ of \(\mathbb S\) is a basis for \(\mathbb H\).

Procedure for Finding the Basis

Step 1

Write the equation

\[c_1\vec{v_1} + c_2 \vec{v_2} + \ldots + c_n \vec{v_n} = \vec 0\]

Step 2

Construct an augmented matrix⁴ associated with the homogeneous system⁵ and transform it to reduced row echelon form.⁶

Step 3

The vectors³ corresponding to the columns containing the leading \(1\)'s form a basis for \(\mathbb W = Span(\mathbb S)\).

References

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1. Read more about sets. ↩↩↩↩↩↩↩↩

2. Read more about vectors. ↩↩↩↩↩

3. Read more about vector spaces. ↩↩↩↩

4. Read more about augmented matrices. ↩

5. Read more about homogeneous linear system. ↩

6. Read more about row echelon form. ↩