Lecture Notes on Linear Transformations and Determinants

These flashcards cover various concepts from linear transformations, determinants, orientations of bases, and associated properties.

Unit Square

The subset of R² defined by S := {α1⃗e1 + α2⃗e2 : 0 ≤ α1, α2 ≤ 1}.

Linear Transformation

A function between two vector spaces that preserves vector addition and scalar multiplication.

Positively Oriented Basis

An ordered basis { ⃗b1, ⃗b2} for R² that can be obtained by rotating ⃗b1 counterclockwise to reach ⃗b2 without crossing the line spanned by ⃗b2.

Negatively Oriented Basis

An ordered basis for R² that cannot be obtained by a counterclockwise rotation from its first vector to its second.

Determinant

The oriented area of the image of a linear transformation, which can indicate preservation or reversal of orientation.

Determinant Formula for 2x2 Matrix

For a 2 x 2 matrix A = [[a, b], [c, d]], the determinant is given by det(A) = ad - bc.

Cofactor Expansion

A method to calculate the determinant of a matrix using minors and cofactor signs.

Parallelogram Area

The area of a parallelogram is calculated as the product of the base length and height.

Bijective Transformation

A transformation that is both injective and surjective, meaning it is one-to-one and onto.

Unit Cube

The subset of R³ defined by C := {α1⃗e1 + α2⃗e2 + α3⃗e3 : 0 ≤ α1, α2, α3 ≤ 1}.

Right-Hand Rule

A method to determine the orientation of a basis in R³ by using the orientation of the index, middle, and thumb fingers.

Volume of a Parallelepiped

The volume can be calculated using the determinant of a matrix defined by its three edges.