9.2 Linear Independence and Bases

Linear independence and bases are crucial concepts in vector spaces. They help us understand how vectors relate to each other and form the foundation of a space.

These ideas are essential for solving systems of equations, analyzing transformations, and working with abstract vector spaces. They connect to the broader study of linear algebra by providing tools to describe and manipulate vector spaces efficiently.

Linear Independence vs Dependence

Defining Linear Independence and Dependence

• A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the other vectors in the set
• A set of vectors is linearly dependent if at least one vector in the set can be written as a linear combination of the other vectors in the set
• The zero vector is always linearly dependent on any set of vectors
• A set containing the zero vector is always linearly dependent ($v_1, v_2, ..., v_n, 0$)
• The trivial solution (all coefficients being zero) is the only solution for a linearly independent set of vectors when written as a linear combination equal to the zero vector ($c_1v_1 + c_2v_2 + ... + c_nv_n = 0$, where $c_1 = c_2 = ... = c_n = 0$)

Relationship between Vector Set Size and Vector Space Dimension

• The number of vectors in a linearly independent set cannot exceed the dimension of the vector space
  • In $\mathbb{R}^2$, a linearly independent set can have at most 2 vectors
  • In $\mathbb{R}^3$, a linearly independent set can have at most 3 vectors
• A set of vectors is linearly dependent if the number of vectors in the set is greater than the dimension of the vector space
  • In $\mathbb{R}^2$, a set of 3 or more vectors is always linearly dependent
  • In $\mathbb{R}^3$, a set of 4 or more vectors is always linearly dependent

Identifying Linear Independence

Setting up and Solving Vector Equations

• To determine if a set of vectors is linearly independent or dependent, set up a vector equation where the linear combination of the vectors is equal to the zero vector and solve for the coefficients ($c_1v_1 + c_2v_2 + ... + c_nv_n = 0$)
• If the only solution to the vector equation is the trivial solution (all coefficients being zero), then the set of vectors is linearly independent
• If there exists a non-trivial solution to the vector equation (at least one coefficient is non-zero), then the set of vectors is linearly dependent

Examples of Linearly Independent and Dependent Sets

• Linearly independent set in $\mathbb{R}^2$: $\{(1, 0), (0, 1)\}$
  • No vector can be written as a linear combination of the other
• Linearly dependent set in $\mathbb{R}^2$: $\{(1, 2), (2, 4), (3, 6)\}$
  • $(3, 6)$ can be written as a linear combination of $(1, 2)$ and $(2, 4)$: $(3, 6) = 1(1, 2) + 1(2, 4)$
• Linearly independent set in $\mathbb{R}^3$: $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$
  • No vector can be written as a linear combination of the others
• Linearly dependent set in $\mathbb{R}^3$: $\{(1, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 1)\}$
  • $(1, 1, 0)$ can be written as a linear combination of $(1, 0, 0)$ and $(0, 1, 0)$: $(1, 1, 0) = 1(1, 0, 0) + 1(0, 1, 0)$

Bases of Vector Spaces

Defining and Identifying Bases

• A basis of a vector space is a linearly independent set of vectors that spans the entire vector space
• Every vector in the vector space can be uniquely expressed as a linear combination of the basis vectors
• The number of vectors in a basis is equal to the dimension of the vector space
  • A basis for $\mathbb{R}^2$ consists of 2 linearly independent vectors
  • A basis for $\mathbb{R}^3$ consists of 3 linearly independent vectors
• In an n-dimensional vector space, any linearly independent set of n vectors forms a basis for the vector space

Standard Basis and Examples

• The standard basis for $\mathbb{R}^n$ is the set of unit vectors $\{e_1, e_2, ..., e_n\}$, where $e_i$ has a 1 in the i-th component and zeros elsewhere
  • Standard basis for $\mathbb{R}^2$: $\{(1, 0), (0, 1)\}$
  • Standard basis for $\mathbb{R}^3$: $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$
• Other examples of bases in $\mathbb{R}^2$:
  • $\{(1, 1), (1, -1)\}$
  • $\{(2, 1), (-1, 1)\}$
• Other examples of bases in $\mathbb{R}^3$:
  • $\{(1, 1, 0), (0, 1, 1), (1, 0, 1)\}$
  • $\{(1, 2, 3), (4, 5, 6), (7, 8, 9)\}$, as long as they are linearly independent

Vector Coordinates in a Basis

Finding Coordinates with respect to a Basis

• The coordinates of a vector with respect to a basis are the coefficients of the linear combination of basis vectors that equals the given vector
• To find the coordinates of a vector with respect to a given basis, set up a vector equation where the linear combination of basis vectors equals the given vector and solve for the coefficients ($c_1v_1 + c_2v_2 + ... + c_nv_n = v$, solve for $c_1, c_2, ..., c_n$)
• The resulting coefficients are the coordinates of the vector with respect to the given basis
• The coordinates of a vector with respect to the standard basis are the components of the vector itself

Changing Basis and Coordinates

• Changing the basis of a vector space results in a change of coordinates for the vectors in the space
• Example: In $\mathbb{R}^2$, the vector $(3, 4)$ has coordinates $(3, 4)$ with respect to the standard basis $\{(1, 0), (0, 1)\}$
  • If we change the basis to $\{(1, 1), (1, -1)\}$, the coordinates of $(3, 4)$ become $(3.5, 0.5)$, since $3.5(1, 1) + 0.5(1, -1) = (3, 4)$
• Example: In $\mathbb{R}^3$, the vector $(2, 3, 4)$ has coordinates $(2, 3, 4)$ with respect to the standard basis $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$
  • If we change the basis to $\{(1, 1, 0), (0, 1, 1), (1, 0, 1)\}$, the coordinates of $(2, 3, 4)$ become $(1, 2, 1)$, since $1(1, 1, 0) + 2(0, 1, 1) + 1(1, 0, 1) = (2, 3, 4)$