Study Guide & Educational Overview
### Linear Algebra Fundamentals:
* **Vector Operations**: The dot product measures collinearity and orthogonality (u · v = 0 means orthogonal); the cross product in R^3 yields a perpendicular vector whose magnitude equals spanned area.
* **Determinants & Invertibility**: A square matrix A is invertible if and only if det(A) ≠ 0; det(A) represents the signed area or volume scaling factor of the transformation.
* **Eigenvalues & Eigenvectors**: Satisfy A*v = λ*v; eigenvectors define directions where the transformation acts purely as scalar stretching or compression.
* **Fundamental Theorems**: Rank-Nullity Theorem states that Rank(A) + Nullity(A) = n (number of columns); Gaussian elimination row-reduces matrices to solve linear systems.
* **Vector Operations**: The dot product measures collinearity and orthogonality (u · v = 0 means orthogonal); the cross product in R^3 yields a perpendicular vector whose magnitude equals spanned area.
* **Determinants & Invertibility**: A square matrix A is invertible if and only if det(A) ≠ 0; det(A) represents the signed area or volume scaling factor of the transformation.
* **Eigenvalues & Eigenvectors**: Satisfy A*v = λ*v; eigenvectors define directions where the transformation acts purely as scalar stretching or compression.
* **Fundamental Theorems**: Rank-Nullity Theorem states that Rank(A) + Nullity(A) = n (number of columns); Gaussian elimination row-reduces matrices to solve linear systems.
Frequently Asked Questions
A determinant of 0 indicates that the matrix squashes space into a lower dimension, has linearly dependent columns, and possesses no multiplicative inverse.
In general, no. AB rarely equals BA even for square matrices of the same dimension.
