This course, which is mostly about algebraic ideas and complements the material in Calculus 1, introduces and develops concepts (including complex numbers and the algebra of polynomials) necessary for a first course in linear algebra. Linear algebra is a subject that grew out of the business of solving systems of linear equations, and then found many other applications in all branches of mathematics and in every science. As this is a first course in linear algebra, our goal is the elementary theory of matrices and determinants, and their application to solving systems of equations, leaving the abstract theory of linear spaces to Linear Mathematics 2. Along the way we include some of the mathematical areas where linear structure emerges as the common feature, to give motivation and illustration for the theory.
|COMPLEX NUMBERS AND POLYNOMIALS. Definition and manipulation of complex numbers; statement of the fundamental theorem of algebra and complete factorization over $\Bbb C$; complex conjugate roots of real polynomials; solution of quadratic and cubic polynomials by radicals.||9|
|MOTIVATIONAL MATERIAL FOR MATRICES AND LINEAR ALGEBRA. Linear ordinary differential equations: general solution of homogeneous linear ODEs with constant coefficients, solution of initial and boundary value problems; examples, including the second order damped harmonic oscillator equation; inhomogeneous equations, resonance.||6|
|VECTORS. Real and complex vectors in $n$ dimensions, addition and scalar multiplication, linear independence, bases for $\Bbb R^n$ and $\Bbb C^n$, the scalar product of two vectors; geometrical representation of real vectors and the scalar product in $\Bbb R^2$ and $\Bbb R^3$, the vector product and triple products in $\Bbb R^3$ , simple applications to geometry and mechanics.||9|
|THEORY OF MATRICES, DETERMINANTS AND LINEAR ALGEBRA. Matrices and systems of linear equations: formulation of systems of linear equations (over $\Bbb R$ and $\Bbb C$) in terms of matrices and vectors; matrix algebra, row and column operations, applications to linear equations, the rank of a matrix, the inverse of a matrix. Determinants and properties of determinants.||15|