3.2 Understand spanning sets and linear dependence

Linear combinations, spanning sets, and linear dependence are key concepts in vector spaces. They help us understand how vectors relate to each other and form the building blocks of larger structures.

These ideas are crucial for grasping the structure of vector spaces. By learning about linear combinations and spanning sets, you'll be able to see how a small set of vectors can generate an entire space.

Linear combinations in vector spaces

Definition and properties of linear combinations

• A linear combination of vectors is a sum of scalar multiples of those vectors
  • If $v_1, v_2, ..., v_n$ are vectors and $c_1, c_2, ..., c_n$ are scalars, then $c_1v_1 + c_2v_2 + ... + c_nv_n$ is a linear combination of the vectors
  • Example: If $v_1 = (1, 2)$ and $v_2 = (3, 4)$, then $2v_1 + 3v_2 = 2(1, 2) + 3(3, 4) = (11, 16)$ is a linear combination of $v_1$ and $v_2$
• The set of all possible linear combinations of a given set of vectors forms a subspace of the vector space, called the span of those vectors
  • The span is the smallest subspace containing all the vectors in the set
  • Example: The span of $\{(1, 0), (0, 1)\}$ in $\mathbb{R}^2$ is the entire $\mathbb{R}^2$ space

Spanning sets and vector spaces

• A set of vectors $\{v_1, v_2, ..., v_n\}$ spans a vector space $V$ if every vector in $V$ can be expressed as a linear combination of $v_1, v_2, ..., v_n$
  • If the span of a set of vectors equals the entire vector space, the set is said to span the vector space
  • Example: The set $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ spans $\mathbb{R}^3$ because any vector $(a, b, c)$ in $\mathbb{R}^3$ can be written as $a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1)$
• The zero vector is always in the span of any set of vectors
• If a set of vectors spans a vector space, adding more vectors to the set will not change the span

Spanning sets of vectors

Determining if a set of vectors spans a vector space

• To determine if a set of vectors spans a vector space, create a matrix $A$ with the given vectors as columns and compute the reduced row echelon form (RREF) of $A$
  • If the RREF has a pivot in every row, the set spans the vector space
  • Example: To check if $\{(1, 1, 1), (1, 0, 2), (2, 1, 3)\}$ spans $\mathbb{R}^3$, form the matrix $A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 1 & 2 & 3 \end{bmatrix}$ and find its RREF. Since the RREF has a pivot in every row, the set spans $\mathbb{R}^3$.
• If a set of vectors does not span a vector space, it is possible to find a vector in the space that cannot be expressed as a linear combination of the given vectors
  • This vector is said to be outside the span of the set
  • Example: The set $\{(1, 0), (0, 1)\}$ does not span $\mathbb{R}^3$ because the vector $(0, 0, 1)$ cannot be expressed as a linear combination of the given vectors

Properties of spanning sets

• If a set of vectors spans a vector space, any superset of that set also spans the vector space
  • Adding more vectors to a spanning set does not change the span
  • Example: If $\{(1, 0), (0, 1)\}$ spans $\mathbb{R}^2$, then $\{(1, 0), (0, 1), (1, 1)\}$ also spans $\mathbb{R}^2$
• If a set of vectors does not span a vector space, any subset of that set also does not span the vector space
  • Removing vectors from a non-spanning set cannot make it span the space
  • Example: If $\{(1, 0, 0), (0, 1, 0)\}$ does not span $\mathbb{R}^3$, then $\{(1, 0, 0)\}$ also does not span $\mathbb{R}^3$

Linear dependence and independence

Definition and properties of linear dependence

• A set of vectors $\{v_1, v_2, ..., v_n\}$ is linearly dependent if there exist scalars $c_1, c_2, ..., c_n$, not all zero, such that $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$
  • In other words, one of the vectors can be expressed as a linear combination of the others
  • Example: The set $\{(1, 2), (2, 4), (3, 6)\}$ is linearly dependent because $(3, 6) = 3(1, 2)$
• If a set of vectors is linearly dependent, any subset of those vectors that includes all the vectors involved in the linear dependence relation is also linearly dependent
  • Example: If $\{v_1, v_2, v_3\}$ is linearly dependent and $v_3 = 2v_1 - v_2$, then $\{v_1, v_2\}$ is also linearly dependent
• Any set of vectors that includes the zero vector is linearly dependent
  • The zero vector can always be expressed as a linear combination of the other vectors with coefficients equal to zero
  • Example: $\{(1, 0), (0, 1), (0, 0)\}$ is linearly dependent because $0(1, 0) + 0(0, 1) + 1(0, 0) = (0, 0)$

Definition and properties of linear independence

• A set of vectors is linearly independent if it is not linearly dependent, i.e., the only solution to $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ is $c_1 = c_2 = ... = c_n = 0$
  • No vector in the set can be expressed as a linear combination of the others
  • Example: The set $\{(1, 0), (0, 1)\}$ is linearly independent because the equation $c_1(1, 0) + c_2(0, 1) = (0, 0)$ has only the trivial solution $c_1 = c_2 = 0$
• To determine if a set of vectors is linearly independent, create a matrix $A$ with the given vectors as columns and solve the homogeneous equation $Ax = 0$
  • If the only solution is the trivial solution ($x = 0$), the set is linearly independent
  • Example: To check if $\{(1, 1, 1), (1, 0, 2), (2, 1, 3)\}$ is linearly independent, solve the equation $\begin{bmatrix} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 1 & 2 & 3 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$. Since the only solution is $x_1 = x_2 = x_3 = 0$, the set is linearly independent.
• If a set of vectors is linearly independent, any subset of those vectors is also linearly independent
  • Removing vectors from a linearly independent set cannot make it linearly dependent
  • Example: If $\{v_1, v_2, v_3\}$ is linearly independent, then $\{v_1, v_2\}$ is also linearly independent

Proofs for spanning sets and dependence

Techniques for proving statements

• To prove a statement about spanning sets or linear dependence, use the definitions and properties of linear combinations, spanning sets, and linear dependence/independence
• Common proof techniques include:
  • Direct proof: Assume the hypothesis and use definitions and properties to derive the conclusion
  • Contradiction: Assume the negation of the conclusion and show that it leads to a contradiction with the hypothesis or known facts
  • Induction: Prove the statement for a base case and then show that if it holds for $n$, it also holds for $n+1$

Examples of statements to prove

• If a set of vectors spans a vector space, any subset of those vectors that includes a linearly dependent set does not span the vector space
  • Proof idea: Use contradiction. Assume the subset spans the space and show that it leads to a contradiction with the linear dependence of the subset.
• If a set of vectors $\{v_1, v_2, ..., v_n\}$ spans a vector space $V$ and $v_n$ is a linear combination of the other vectors, then $\{v_1, v_2, ..., v_{n-1}\}$ also spans $V$
  • Proof idea: Use direct proof. Express any vector in $V$ as a linear combination of $\{v_1, v_2, ..., v_n\}$ and then substitute the linear combination of $v_n$ in terms of the other vectors.
• If a set of vectors is linearly independent, any subset of those vectors is also linearly independent
  • Proof idea: Use contradiction. Assume the subset is linearly dependent and show that it leads to a contradiction with the linear independence of the original set.