Span: Imagine you have a set of vectors. The "span" of these vectors is all the points (or vectors) you can possibly reach by just stretching, shrinking, and adding your starting vectors together (linear combinations). It's the geometric space (like a line, a plane, or the entire 3D space) generated by those vectors.
Basis: A "basis" for a vector space is like a minimal set of building block vectors for that space. It's a small, efficient set of vectors that are independent (no redundancy - linearly independent) and whose span covers the entire space. Think of them as the fundamental coordinate directions for that space.
The span of a set of vectors \(S = \{\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_k\}\) is the set of all possible linear combinations of these vectors.
$$ \text{Span}(S) = \left{ c_1\mathbf{v}1 + c_2\mathbf{v}_2 + \dots + c_k\mathbf{v}_k \mid c_1, ..., c_k \in \mathbb{R} \right} = \left{ \sum \right} $$}^k c_i \mathbf{v}_i \mid c_i \in \mathbb{R
The span forms a Vector Space (or a subspace of a larger vector space).
Spanning: The vectors span the entire space \(V\) (\(\text{Span}(B) = V\)).
Dimension: The number of vectors (\(n\)) in any basis for a vector space \(V\) is always the same. This number \(n\) is called the dimension of the vector space \(V\), denoted \(\dim(V)\).
Span of one non-zero vector: A line through the origin in the direction of the vector.
Span of two linearly independent vectors: A plane through the origin containing both vectors.
Span of three linearly independent vectors in 3D space: The entire 3D space (\(\mathbb{R}^3\)).
If vectors are linearly dependent, adding a dependent vector does not increase the dimension of the span (e.g., the span of two vectors lying on the same line is still just that line).
Minimality: A basis is a minimal set of vectors that spans the space (removing any vector would make it no longer span the space).
Maximality: A basis is a maximal set of linearly independent vectors within the space (adding any other vector from the space would make the set linearly dependent).
Unique Representation: Every vector \(\mathbf{v}\) in the space \(V\) can be written as a unique linear combination of the basis vectors \(\{\mathbf{b}_1, ..., \mathbf{b}_n\}\). The coefficients of this combination, \(c_1, ..., c_n\), are the coordinates of \(\mathbf{v}\) relative to that basis \(B\). \(\mathbf{v} = \sum c_i \mathbf{b}_i\).
The most common basis for \(\mathbb{R}^n\) is the standard basis (or natural basis), consisting of vectors \(\mathbf{e}_i\) with a single 1 in the \(i\)-th position and 0s elsewhere.
For \(\mathbb{R}^2\): \(\mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\), \(\mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\). (Vectors along the x and y axes).
For \(\mathbb{R}^3\): \(\mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\), \(\mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\), \(\mathbf{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\). (Vectors along x, y, z axes).
Column Space (Col(A)): The span of the columns of a matrix\(\mathbf{A}\). It's the set of all possible outputs \(\mathbf{Ax}\). The dimension of the column space is the rank of the matrix. A basis for Col(A) can be found from the pivot columns of the original matrix \(\mathbf{A}\).
Row Space (Row(A)): The span of the rows of a matrix \(\mathbf{A}\). Its dimension also equals the rank. A basis for Row(A) can be found from the non-zero rows of the row-echelon form of \(\mathbf{A}\).
Span: The set of all reachable vectors via linear combinations of a starting set. Forms a vector space (line, plane, etc.). \(\text{Span}(S) = \{ \sum c_i \mathbf{v}_i \}\).
Basis: A set of vectors \(B\) that is both linearly independent and spans the entire vector space (\(\text{Span}(B)=V\)).
A basis provides a minimal set of "building blocks" or "coordinate directions" for the space.
The number of vectors in a basis defines the dimension of the space.
The standard basis (\(\mathbf{e}_1, \mathbf{e}_2, ...\)) is the simplest basis for \(\mathbb{R}^n\).