If a and B Are N by N Matrices with Inverses, (AB)-1=B-1A-1

Math 100 Exam 2 Formulas

If u and v are vectors in 2-space or 3-space and θ is the angle between them, then the dot product or inner product u.v is defined by

Distance from point to a line in 2-space:

If and are vectors in 3-space, then the cross product is the vector or in determinant notation

The absolute value of the determinant is equal to the area of the parallelogram in 2-space determined by the vectors

The absolute value of the determinant is equal to the volume of the parallelepiped in 3-space determined by the vectors

Volume of parallelepiped determined by vectors u, v, and w is given by

Point normal form of the equation of a plane: a(x-x0)+b(y-y0)+c(z-z0)=0

Equation of a plane with {a,b,c} as a normal vector: ax+by+cz+d=0

Parametric equations of a line:

Distance from a point to a plane in 3-space:

If vectors are written as column matrices,

Matrix notation for a linear transformation from n-space to m-space:

Standard Matrix for Reflection about y-axis in 2-space:

Standard Matrix for Reflection about x-axis in 2-space:

Standard Matrix for Reflection about the line x=y in 2-space:

Standard Matrix for Reflection about the xy-plane in 3-space:

Standard Matrix for Reflection about the xz-plane in 3-space:

Standard Matrix for Reflection about the yz-plane in 3-space:

Standard Matrix for Orthogonal projection on the x-axis in 2-space:

Standard Matrix for Orthogonal projection on the y-axis in 2-space:

Standard Matrix for the Orthogonal projection on the xy-plane in 3-space:

Standard Matrix for the Orthogonal projection on the xz-plane in 3-space:

Standard Matrix for the Orthogonal projection on the yz-plane in 3-space:

Standard Matrix for the rotation through angle θ in 2-space:

Standard Matrix for the rotation through angle θ about the positive x-axis in 3-space:

Standard Matrix for the rotation through angle θ about the positive y-axis in 3-space:

Standard Matrix for the rotation through angle θ about the positive z-axis in 3-space:

Standard Matrix for Contraction or dilation by factor k in 2-space:

Standard Matrix for Contraction or dilation by factor k in 3-space:

A linear transformation is one-to-one if T maps distinct vectors (points) in into distinct vectors (points) in

Equivalent statements: If A is an n by n matrix and is multiplication by A, then the following statements are equivalent: (a) A is invertible; (b) The range of is ; (c) is one-to-one

A transformation is linear if and only if the following relationships hold for all vectors u and v inand every scalar c: (a) T(u+v)=T(u)+T(v) and

(b) T(cu)=cT(u)