Spanning Sets and Linear Combinations in Linear Algebra

If the span of a set $S$ covers every single vector in $\mathbb{R}^n$, we say that $S$ is a spanning set (or a generating set) for $\mathbb{R}^n$.

The Connection to Linear Systems

There is a deep connection between spanning sets and solving linear systems $Ax = b$. To ask if a vector $b$ is in the span of the columns of matrix $A$ is exactly the same as asking if the system $Ax = b$ is consistent (has a solution). If you can find the scalars (the $x$ values) that satisfy the equation, you have successfully expressed $b$ as a linear combination of the columns.

Linear Independence and Redundancy

While a spanning set tells us if we have enough vectors to cover a space, linear independence tells us if we have too many.

A set of vectors is linearly dependent if there is redundancy in the set—meaning at least one vector can be reconstructed using a linear combination of the others. Formally, a set is linearly dependent if there exist scalars $c_1, c_2, \dots, c_k$, not all of which are zero, such that: $c_1v_1 + c_2v_2 + \dots + c_kv_k = 0$ Conversely, a set is linearly independent if the only way to reach the zero vector is to set all scalars to zero.

Exam Tip: The Two-Vector Shortcut To quickly check if exactly two vectors are linearly independent, simply check if they are scalar multiples of one another. If one is not just a scaled version of the other (e.g., $u \neq c \cdot v$), they are linearly independent and point in different directions.

Basis and Dimension

A basis is the "Goldilocks" set of linear algebra: it is a set of vectors that is just right. To qualify as a basis for a subspace, a set must satisfy two strict conditions:

1. They must span the subspace (they can build everything in the set).
2. They must be linearly independent (there is no redundancy).

If you are given a large spanning set and want to find a basis, you can simply "boil it down" by removing the redundant vectors until only the essential directions remain.

Standard Unit Vectors

In $\mathbb{R}^n$, the most common basis is the standard basis, consisting of the standard unit vectors $e_1, e_2, \dots, e_n$. For example, in $\mathbb{R}^3$: $e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ These vectors lie along the coordinate axes and can be used to build any vector in the space through simple scaling.

Matrix Subspaces

Every matrix $A$ is associated with three fundamental subspaces that describe its structure:

• Row Space: The span of the rows of $A$. If two matrices are row equivalent (meaning one can be transformed into the other via row operations), they share the exact same row space. A practical way to find a basis for the row space is to take the non-zero rows of the reduced row echelon form (RREF) of the matrix.
• Column Space: The span of the columns of $A$. While row operations change the column space, they do not change the row space.
• Null Space: The set of all solutions to the homogeneous system $Ax = 0$. This set is always a subspace because it contains the zero vector and is closed under both vector addition and scalar multiplication.

Properties of Linear Systems

When dealing with a consistent system $Ax = b$, the number of solutions is strictly limited by the structure of the vectors:

• Unique Solution: The system has exactly one point of intersection.
• Infinitely Many Solutions: If there are at least two distinct solutions, there are automatically infinitely many. This happens because you can take the difference between two solutions (which will be a solution to the homogeneous system) and add any scalar multiple of it to your original solution, creating a line, plane, or hyperplane of solutions.