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Mathematics for Physics

13 Linear Algebra

Linear algebra is the study of vector spaces and the linear maps between them.

A vector space is a structured set that is used to add and scale its elements, where an element is called a vector and a scaling number is called a scalar.

Here are the ideas that are fundamental to the study of linear algebra.

Linear combination and span. A linear combination is a vector: the result of scaling given vectors and adding them. The span is a set: every linear combination of a chosen list of vectors.

Basis and dimension. A basis is a list of linearly independent vectors that spans the space. Linear independence is a property: no vector in the list is a combination of the others. Dimension is a number: the length of any basis.

Linear transformations. A linear transformation is a map between vector spaces that preserves addition and scaling. Preservation means the image of a sum is the sum of the images.

definition [d]

A vector space over a field of scalars is a nonempty set \(V\) of vectors with addition and scalar multiplication such that for all \(u,v,w \in V\) and all scalars \(c,d\):

  1. \(u+v = v+u\)
  2. \((u+v)+w = u+(v+w)\)
  3. there is a zero vector \(0\) with \(u+0 = u\)
  4. each \(u\) has an additive inverse \(-u\) with \(u+(-u) = 0\)
  5. \(c(du) = (cd)u\)
  6. \(c(u+v) = cu+cv\)
  7. \((c+d)u = cu+du\)
  8. \(1u = u\)

where

  • \(V\) is the set of vectors
  • \(c,d\) are scalars
  • \(u,v,w\) are vectors in \(V\)

Note:

  • The two operations are vector addition and scalar multiplication.
  • The same axioms define a vector space for any field of scalars.

13.1 Examples

13.1.1 Simple

Linear algebra studies vectors and linear maps. Here three vectors in a plane.

\[ V = \{ e_{1},\ e_{2},\ v \} \]

\[ e_{1} = (1,0),\quad e_{2} = (0,1),\quad v = (2,3) \]

where

  • \(V\) is a set of sample vectors.
  • \(e_{1}, e_{2}\) form the standard basis of \(\mathbb{R}^{2}\).

13.1.2 General

Matrices represent linear maps. A \(3 \times 3\) matrix acts on \(\mathbb{R}^{3}\).

\[ A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} \]

\[ A(x_{1},x_{2},x_{3}) = (x_{1},\ 2 x_{2},\ 3 x_{3}) \]

where

  • \(A\) is a linear transformation written as a matrix.

13.2 References

  1. Axler, S. Linear Algebra Done Right. Springer, 2015. — vector spaces and the linear maps between them; linear combinations and span.
  2. Shilov, G. E. Linear Algebra. Dover, 1977. — basis and dimension.
  3. Shifrin, T., & Adams, M. Linear Algebra: A Geometric Approach. — linear transformations as maps that preserve addition and scaling.
  4. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1978. — the axioms of a vector space.