The span of a set of vectors is a fundamental concept in linear algebra, referring to all possible linear combinations of those vectors. To break this down further:

1. Definition:

If you have a set of vectors $v_1, v_2, \dots, v_n$ in a vector space (like $\mathbb{R}^n$ or $\mathbb{C}^n$), the span of these vectors is the collection of all vectors that can be written as a linear combination of them. Formally, the span of vectors $v_1, v_2, \dots, v_n$ is:

$\text{span}\{v_1, v_2, \dots, v_n\} = \left\{ c_1 v_1 + c_2 v_2 + \dots + c_n v_n \mid c_1, c_2, \dots, c_n \in \mathbb{R} \right\}$

Where $c_1, c_2, \dots, c_n$ are scalars (real or complex numbers).

2. Key Points:

• The span is a vector space itself, meaning it includes the zero vector and is closed under vector addition and scalar multiplication.
• If the span of a set of vectors covers the whole space (e.g., $\mathbb{R}^3$), we say these vectors span the space.
• The span represents the subspace of the vector space that the set of vectors can "reach" through their linear combinations.

3. Example:

Consider two vectors $v_1 = (1, 0)$ and $v_2 = (0, 1)$ in $\mathbb{R}^2$. Their span is:

$\text{span}\{(1,0), (0,1)\} = \left\{ c_1(1,0) + c_2(0,1) \mid c_1, c_2 \in \mathbb{R} \right\} = \mathbb{R}^2$

In this case, any vector in $\mathbb{R}^2$ can be written as a linear combination of $v_1$ and $v_2$, so they span the whole plane.

4. Geometric Interpretation:

• In $\mathbb{R}^2$, the span of a single vector $v_1$ forms a line through the origin.
• The span of two non-parallel vectors in $\mathbb{R}^2$ forms the entire plane.
• In $\mathbb{R}^3$, the span of two vectors that are not collinear forms a plane, and three linearly independent vectors span the entire 3D space.

5. Linear Independence and Span:

If the vectors in a set are linearly independent, the span of those vectors is the smallest vector space that contains them. If they are linearly dependent, the span is a subspace of lower dimension, because some vectors in the set can be written as combinations of others.

Example Problem:

If $v_1 = (1, 2)$ and $v_2 = (3, 6)$, then the span of $\{v_1, v_2\}$ is not all of $\mathbb{R}^2$, because $v_2$ is a scalar multiple of $v_1$, meaning they lie on the same line. So their span is a line, not a plane.

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Would you like further clarification on any aspect of this?

Related Questions:

1. How does the span relate to the concept of a basis?
2. What is the difference between the span and the null space?
3. How can you find the dimension of the span of a set of vectors?
4. What happens if all vectors in the set are linearly dependent?
5. How does the span change in different vector spaces (e.g., real vs. complex spaces)?

Tip:

To determine if a set of vectors spans a space, check if they are linearly independent. If they are, they span the space with a dimension equal to the number of vectors.