1.3 Linear Combinations and Linear Independence

Linear combinations and independence are key concepts in vector spaces. They help us understand how vectors relate to each other and form subspaces. These ideas are crucial for grasping the structure of vector spaces and their subspaces.

By exploring linear combinations, we can see how vectors can be built from others. Linear independence shows which vectors are truly unique. These concepts lay the groundwork for understanding bases and dimensions in vector spaces.

Linear Combinations of Vectors

Definition and Properties

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• A linear combination of vectors is a sum of scalar multiples of the vectors
  • For vectors $v_1, v_2, ..., v_n$ in a vector space $V$ and scalars $c_1, c_2, ..., c_n$ in a field $F$, a linear combination is $c_1v_1 + c_2v_2 + ... + c_nv_n$
  • The scalars in a linear combination are called the coefficients
• The set of all possible linear combinations of a given set of vectors forms a subspace of the vector space
• If the zero vector can be expressed as a linear combination of a set of vectors with not all coefficients being zero, then the set is linearly dependent

Examples and Applications

• A linear combination of two vectors $v_1 = (1, 2)$ and $v_2 = (3, 4)$ with coefficients $c_1 = 2$ and $c_2 = -1$ is $2v_1 - v_2 = (2, 4) - (3, 4) = (-1, 0)$
• In physics, the resultant force acting on an object can be expressed as a linear combination of the individual forces acting on it
• In computer graphics, points on a 3D surface can be represented as linear combinations of the vertices defining the surface

Expressing Vectors as Linear Combinations

Solving for Coefficients

• To determine if a vector $v$ can be expressed as a linear combination of vectors $v_1, v_2, ..., v_n$, solve the equation $c_1v_1 + c_2v_2 + ... + c_nv_n = v$ for the coefficients $c_1, c_2, ..., c_n$
  • If a solution exists for the coefficients, then $v$ is a linear combination of the given vectors
  • If the only solution for the coefficients is the trivial solution (all coefficients are zero), then $v$ is not a linear combination of the given vectors
• The process of solving for the coefficients involves setting up a system of linear equations and using techniques such as Gaussian elimination or matrix inversion

Examples and Applications

• To express the vector $(2, 3)$ as a linear combination of the vectors $(1, 1)$ and $(1, 2)$, solve the equation $c_1(1, 1) + c_2(1, 2) = (2, 3)$ for $c_1$ and $c_2$. The solution is $c_1 = 1$ and $c_2 = 1$, so $(2, 3) = 1(1, 1) + 1(1, 2)$
• In cryptography, a message can be encoded as a linear combination of basis vectors, and decoding involves expressing the encoded message as a linear combination of the same basis vectors

Linear Independence and Dependence

Definitions and Properties

• A set of vectors is linearly independent if no vector in the set can be expressed 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 expressed as a linear combination of the other vectors in the set
  • For a set of vectors $v_1, v_2, ..., v_n$, the set is linearly dependent if there exist scalars $c_1, c_2, ..., c_n$, not all zero, such that $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$
  • If the only solution to the equation $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ is the trivial solution (all coefficients are zero), then the set of vectors is linearly independent
• The zero vector is always linearly dependent on any set of vectors
• A set containing the zero vector is always linearly dependent

Examples and Applications

• The set of vectors $\{(1, 0), (0, 1)\}$ is linearly independent in $\mathbb{R}^2$, as neither vector can be expressed as a scalar multiple of the other
• The set of vectors $\{(1, 2), (2, 4), (3, 6)\}$ is linearly dependent, as $(3, 6) = 1(1, 2) + 1(2, 4)$
• In quantum mechanics, a set of quantum states is linearly independent if no state can be expressed as a linear combination of the others

Identifying Linear Independence vs Dependence

Solving the Homogeneous Equation

• To determine if a set of vectors $v_1, v_2, ..., v_n$ is linearly independent or dependent, solve the equation $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ for the coefficients $c_1, c_2, ..., c_n$
  • If the only solution is the trivial solution (all coefficients are zero), then the set is linearly independent
  • If there exists a non-trivial solution (at least one coefficient is non-zero), then the set is linearly dependent
• The process of solving for the coefficients involves setting up a system of linear equations and using techniques such as Gaussian elimination or matrix inversion

Additional Considerations

• If the number of vectors in the set is greater than the dimension of the vector space, then the set is necessarily linearly dependent
• If a subset of a linearly independent set is removed, the remaining set is still linearly independent
• In $\mathbb{R}^n$, a set of $n$ vectors is linearly independent if and only if the determinant of the matrix formed by the vectors as columns is non-zero