Linear Algebra Through Geometry

3-space a₁ a₂ ax² b₁ b₂ b3 b₁x₁ calculate Chapter 2.5 characteristic equation column coordinate axis define det(m determinant diagonal matrix dimensions dot product E₁ E₂ eigenvalues eigenvector corresponding eigenvectors elementary matrices EXAMPLE Exercise 11 Exercise 9 F₁ Figure Find all solutions formula functions geometry Hence homogeneous system illus inverse isometry length linear algebra linear transformation linearly dependent nonzero vector obtained origin orthogonal matrix pair of vectors parallelogram perpendicular plane positively oriented preserves orientation PROOF Proposition quadric surface radians real number reflection rotation satisfies scalar multiple segment Show Similarly subspace symmetric matrix Systems of Equations T(X₁ t₁ t₂ Theorem transformation of R3 transformation with matrix triplet u₁ unit vector vectors in R4 x-axis X₁ X₂ Y₁ zero а2 аз