5.1 Definition and properties of multivectors

Multivectors are the Swiss Army knives of geometric algebra. They combine different grades of blades, letting us represent complex geometric objects with a single mathematical entity. From simple scalars to intricate higher-dimensional constructs, multivectors cover it all.

Understanding multivectors is key to mastering geometric algebra. They form a vector space, allowing us to add them together and multiply them by scalars. This flexibility makes multivectors powerful tools for solving geometric problems and representing transformations in various dimensions.

Multivectors as linear combinations

Definition and general form

• A multivector is a sum of blades with scalar coefficients, where blades are the basic building blocks of geometric algebra
• The general form of a multivector is $A = \sum a_i e_i$, where $a_i$ are scalars and $e_i$ are blades of different grades
  • Example: $A = 2 + 3e_1 + 4e_2 + 5e_1 \wedge e_2$, where $1$, $e_1$, $e_2$, and $e_1 \wedge e_2$ are blades of grades 0, 1, 1, and 2, respectively

Homogeneous and inhomogeneous multivectors

• Multivectors can be homogeneous (consisting of blades of the same grade) or inhomogeneous (consisting of blades of different grades)
  • Example of a homogeneous multivector: $A = 2e_1 + 3e_2$, where both blades have grade 1
  • Example of an inhomogeneous multivector: $B = 1 + 4e_1 + 5e_1 \wedge e_2$, where blades have grades 0, 1, and 2
• The grade of a multivector is the highest grade of its constituent blades
  • Example: The grade of $B = 1 + 4e_1 + 5e_1 \wedge e_2$ is 2, as the highest grade blade is $e_1 \wedge e_2$

Multivectors as a vector space

• Multivectors form a vector space over the field of scalars, typically real numbers
  • This means that multivectors can be added together and multiplied by scalars, following the axioms of a vector space
  • The zero multivector, denoted by $0$, serves as the identity element for addition
  • The negative of a multivector $A$ is defined as $-A$, satisfying $A + (-A) = 0$

Properties of multivectors

Addition of multivectors

• Multivectors can be added together, and the result is another multivector
• Addition of multivectors is commutative: $A + B = B + A$
  • Example: $(2 + 3e_1) + (4e_2 + 5e_1 \wedge e_2) = (4e_2 + 5e_1 \wedge e_2) + (2 + 3e_1)$
• Addition of multivectors is associative: $(A + B) + C = A + (B + C)$
  • Example: $((2 + 3e_1) + (4e_2 + 5e_1 \wedge e_2)) + (6e_1 + 7e_2) = (2 + 3e_1) + ((4e_2 + 5e_1 \wedge e_2) + (6e_1 + 7e_2))$

Scalar multiplication of multivectors

• Scalar multiplication of a multivector $A$ by a scalar $\lambda$ is defined as $\lambda A = \sum (\lambda a_i) e_i$, where $a_i$ are the scalar coefficients of $A$ and $e_i$ are the blades
  • Example: If $A = 2 + 3e_1 + 4e_2$ and $\lambda = 2$, then $\lambda A = 2(2 + 3e_1 + 4e_2) = 4 + 6e_1 + 8e_2$
• Scalar multiplication is distributive over addition of multivectors: $\lambda(A + B) = \lambda A + \lambda B$
  • Example: If $A = 2 + 3e_1$, $B = 4e_2 + 5e_1 \wedge e_2$, and $\lambda = 2$, then $\lambda(A + B) = 2(2 + 3e_1 + 4e_2 + 5e_1 \wedge e_2) = (4 + 6e_1) + (8e_2 + 10e_1 \wedge e_2) = \lambda A + \lambda B$
• Scalar multiplication is compatible with the field of scalars: $(\lambda \mu) A = \lambda (\mu A)$, where $\lambda$ and $\mu$ are scalars
  • Example: If $A = 2 + 3e_1$, $\lambda = 2$, and $\mu = 3$, then $(\lambda \mu) A = (2 \cdot 3)(2 + 3e_1) = 6(2 + 3e_1) = 12 + 18e_1$ and $\lambda (\mu A) = 2(3(2 + 3e_1)) = 2(6 + 9e_1) = 12 + 18e_1$

Operations on multivectors

Addition of multivectors

• To add two multivectors $A = \sum a_i e_i$ and $B = \sum b_i e_i$, add the corresponding scalar coefficients: $A + B = \sum (a_i + b_i) e_i$
  • Example: If $A = 2 + 3e_1$ and $B = 4e_2 + 5e_1 \wedge e_2$, then $A + B = (2 + 0) + (3e_1 + 0) + (0 + 4e_2) + (0 + 5e_1 \wedge e_2) = 2 + 3e_1 + 4e_2 + 5e_1 \wedge e_2$
• The zero multivector, denoted by $0$, is the multivector with all scalar coefficients equal to zero. It serves as the identity element for addition
  • Example: $A + 0 = 0 + A = A$ for any multivector $A$
• The negative of a multivector $A = \sum a_i e_i$ is defined as $-A = \sum (-a_i) e_i$. It satisfies $A + (-A) = 0$
  • Example: If $A = 2 + 3e_1$, then $-A = -2 - 3e_1$, and $A + (-A) = (2 + 3e_1) + (-2 - 3e_1) = 0$

Scalar multiplication of multivectors

• To multiply a multivector $A = \sum a_i e_i$ by a scalar $\lambda$, multiply each scalar coefficient by $\lambda$: $\lambda A = \sum (\lambda a_i) e_i$
  • Example: If $A = 2 + 3e_1 + 4e_2$ and $\lambda = 2$, then $\lambda A = 2(2 + 3e_1 + 4e_2) = 4 + 6e_1 + 8e_2$
• Scalar multiplication is distributive over addition of multivectors: $\lambda(A + B) = \lambda A + \lambda B$
  • Example: If $A = 2 + 3e_1$, $B = 4e_2 + 5e_1 \wedge e_2$, and $\lambda = 2$, then $\lambda(A + B) = 2(2 + 3e_1 + 4e_2 + 5e_1 \wedge e_2) = (4 + 6e_1) + (8e_2 + 10e_1 \wedge e_2) = \lambda A + \lambda B$

Multivectors vs Exterior Algebra

Relationship between multivectors and exterior algebra

• The exterior algebra, also known as the Grassmann algebra, is a subalgebra of the geometric algebra
• Multivectors in the exterior algebra are linear combinations of blades, just like in the geometric algebra
  • Example: In the exterior algebra, a multivector can be expressed as $A = a_0 + a_1 e_1 + a_2 e_2 + a_{12} e_1 \wedge e_2$, where $a_0, a_1, a_2, a_{12}$ are scalars and $1, e_1, e_2, e_1 \wedge e_2$ are blades of grades 0, 1, 1, and 2, respectively
• The exterior algebra is a graded algebra, with the grade of a blade determined by the number of vectors in its exterior product
  • Example: The grade of $e_1 \wedge e_2 \wedge e_3$ is 3, as it is the exterior product of three vectors

Differences between geometric algebra and exterior algebra

• In the exterior algebra, the product of two vectors is their exterior product, which is antisymmetric
  • Example: In the exterior algebra, $e_1 \wedge e_2 = -e_2 \wedge e_1$
• The exterior algebra is generated by the exterior product of vectors, resulting in blades of different grades
  • Example: In the exterior algebra, the basis elements are $1, e_1, e_2, ..., e_n, e_1 \wedge e_2, e_1 \wedge e_3, ..., e_{n-1} \wedge e_n, ..., e_1 \wedge e_2 \wedge ... \wedge e_n$
• The geometric algebra extends the exterior algebra by including the geometric product, which is a combination of the exterior product and the inner product
  • Example: In the geometric algebra, the geometric product of two vectors $a$ and $b$ is given by $ab = a \cdot b + a \wedge b$, where $a \cdot b$ is the inner product and $a \wedge b$ is the exterior product