VECTOR SPACE
4.1 Vectors
Definition and Representation
Vector: A quantity having both magnitude and direction. Represented in 2-space as:
u or AB, starting from point A(a1, a2) to point B(b1, b2).
Notation:
Vectors are usually denoted by lowercase letters (e.g., u, v).
Components of a vector in 2-space: AB = B - A = (b1 - a1, b2 - a2).
Vectors in R³
In 3-space, a vector AB represents:
AB = (b1 - a1, b2 - a2, b3 - a3).
Common vectors in engineering and physics are often represented in either 2-space or 3-space.
Vector Operations
Vector Addition:
To add two vectors u and v in R^2, simply add their corresponding components:
u + v = (u1 + v1, u2 + v2).
Scalar Multiplication:
To multiply a vector u by a scalar c, multiply each component of the vector by c:
cu = (cu1, cu2).
Zero Vector
A vector with all components equal to zero is called the zero vector (denoted 0) and acts as the additive identity.
Properties of Vectors
Equivalence: Two vectors are equivalent if their corresponding components are equal.
Unit Vector: A vector with a magnitude of 1, often found by dividing a vector by its magnitude.
4.1.1 Vectors in a Plane and Space
2D Vectors:
A standard vector can be represented geometrically by originating from the origin O(0, 0).
Components of the vector can also be understood as coordinates of the endpoint.
Basic Operations on Vectors
Vector Addition Example
For vectors u = (1, -2) and v = (3, 5):
u + v = (1 + 3, -2 + 5) = (4, 3).
Scalar Multiplication Example
For vector u = (1, -3) and scalar c = 3:
cu = (31, 3(-3)) = (3, -9).
4.1.2 Dot Product and Applications
Dot Product Definition: If u and v are vectors:
The dot product is defined as: u.v = u1v1 + u2v2.
Angle Correlation: The angle between two vectors can be calculated using:
cos(θ) = (u.v) / (|u| |v|).
Types of Angles
Acute Angle: u.v > 0
Obtuse Angle: u.v < 0
Right Angle: u.v = 0
4.2 Real Vector Spaces
Definition
A vector space is a set of vectors along with two operations: vector addition and scalar multiplication.
Closure properties must be satisfied, meaning that the result of both operations must also be in the same set.
Axioms for Vector Spaces
Axiom 1-10: Various properties dictate how addition, scalar multiplication, the existence of additive identity and inverses function.
Examples of Vector Spaces
R^n, sets of polynomials, and sets of matrices can all form vector spaces if they satisfy the aforementioned properties.
4.3 Subspaces
Definition of Subspace
A subset W of a vector space V is a subspace if it meets:
It is closed under vector addition.
It is closed under scalar multiplication.
Properties to Verify Subspace
Check if the zero vector is present in W.
Examples of Subspaces
Sets such as all columns or rows of a certain matrix can be examples of subspaces.
4.4 Linear Combination, Spanning Set and Linear Independence
Linear Combination
A vector x is a linear combination of vectors if it can be expressed as a weighted sum of those vectors.
Spanning Set
A set of vectors spans vector space V if every vector in V can be expressed as a linear combination of the vectors in the set.
Linear Independence
A set of vectors is linearly independent if the only solution to the equation c₁v₁ + c₂v₂ + ... + cₙvₙ = 0 is c₁ = c₂ = ... = cₙ = 0.
4.5 Basis and Dimension
Basis
A basis for a vector space is a set of vectors that spans the space and is linearly independent.
Dimension
The dimension of a vector space is the number of vectors in any basis of that space, indicating its size.
4.6 Row Space, Column Space, Nullspace, Rank and Nullity
Row Space
The row space of a matrix A is the subspace spanned by its rows.
Column Space
The column space of A is spanned by its columns.
Nullspace
The nullspace consists of all vectors x such that Ax = 0, based on the solutions to the homogeneous equation.
Rank and Nullity
Rank: The dimension of the row space or column space.
Nullity: The dimension of the nullspace.