Chapter 1-2 Formulas/Methods
1.
Solving System of Equations:
Gauss Elimination (Row echelon form)
Gauss-Jordan Elimination (Reduced row echelon)
X = A-1 . b
det( A1 )
etc.
det( A)
Cramer’s Rule X1 =
2.
Finding Inverse of a Matrix:
2 X 2 …………
n X n ………….
1
ad  bc
A-1 =
A-1 =
d b
c a
1
[ Adj ( A) ]
det( A)
n X n ………… [A | I ]  [ I | A-1 ] using elementary row operations
3.
Determinants:
Sum of all signed elementary products.
For 2 X 2 determinant we use |A| = ad – bc
when A =
a
c
b
d
For 3X3 we collect all signed diagonal products ( upon rewriting first two columns)
Otherwise use the method of cofactors:
| A | = a11 C11 + a12 C12 + ……. + a1n C1n, Cij is cofactor of Cij
= a11 M11 - a12 M12 +
+ ( -1)1+ n M1n, Cij = (-1)I+ j Mij
Chapter 3 Formulas (Vectors)
U = u1 i + u2 j + u3 k = ( u1, u2, u3 ) . Here i ,j and k are unit vectors along x, y, and z axis.
If P (x1, y1,z1) and Q (x2,y2,z2) are two points, then the vector PQ = (x2 – x1, y2 –y1, z2 - z1)
||U|| = length of vector U =
u1  u22  u32
2
Dot Product U.V = ||U|| ||V|| cos , where  is the angle between vectors U and V.
cos =
U .V
,
|| U |||| V ||
||Proj a U|| =
U .a
,
|| a ||
(This formula gives the angle between two vectors)
Proj a U =
(
U .a
a
)
=
|| a || || a ||
Orthogonal Projection of U in the direction of a =
U-
U .a
a
|| a ||2
U .a
a
|| a ||2
Distance between a point P(m,n) and a line Ax+By+C = 0 is
d=
| Am  Bm  C |
A2  B 2
Normal vector to line Ax + By + C = 0 is N = ( A, B).
The cross-product of two vectors U and V is U X V =
i
j
U1
V1
k
U2
V2
U3
V3
U X V is perpendicular to U and V. And ||U X V|| = ||U|| ||V|| sin , where  is the angle between U and
V.
The scalar triple product or mixed product of U, V, and W is U. (V X W) = volume of parallelepiped
formed by vectors U, V, and W. If U, V, and W are in same plane, their mixed product will be zero.
Distance of point (m,n,p) from plane Ax+By+Cz + D = 0 is d =
| Am  Bn  Cp  D |
A2  B 2  C 2
Normal to plane Ax+By+Cz+D= 0 is N = (A,B,C)
A line through P(x0, y0, z0) parallel to vector V(a,b,c) has parametric equations of:
x = x0 + a t, y = y0 + b t, z = z0 + c t
Index of Symbols
Chapter 1
[aij], a matrix whose entry in i th row and j th column is aij
[A | b ], augmented matrix of a system of linear equations
m x n, size of a matrix with m rows and n columns
AT, transpose of a matrix
Tr(A), trace of a matrix A
In, n x n identity matrix
A-1, inverse of a square matrix
AT = A, symmetric matrix, AT = - A, skew symmetric matrix
Ax = b, system of linear equations
0, zero matrix
Chapter 2
Det(A), determinant of a square matrix
E, an elementary matrix
Mij, minor of entry aij in a square matrix A
Cij, cofactor of entry aij in a square matrix A
Adj(A), adjoint of a square matrix A
Det( I – A) = 0, characteristic of a square matrix A.
Chapter 3
V, vector
0, zero vector
|| u ||, norm of a vector u
u.v, dot product or inner product of two vectors u x v, cross
product
proja u, vector component of u along a
u – proja u, vector component of u orthogonal to a, i, j, k unit
vectors
u.(v x w), scalar triple product
a(x-x0) + b(y-y0) + c(z-z0) = 0, point normal form of equation to
plane
ax + by + cz + d = 0, general form of the equation of plane
n.(r-r0) = 0, vector form of the equation of plane
Chapter 4
(a1, a2, ……, an) ordered n-tuples in n-space Rn.
T: Rn  Rm or T ( x1, x2, …., xn ) = ( w1, w2, ….., wm ). The transformation T assigns
each point in Rn i.e. ( x1, x2, …., xn ) a point ( w1, w2, ….., wm ) in Rm.
A linear transformation T: Rn  Rm is said to be one-to-one if T maps distinct vectors (points) in
Rn into distinct vectors (points) in Rm.
Properties of linear transformation: T (u + v) = T(u) + T (v), and T (cu) = c T(u).
Chapter 10
i, symbol
to denote
1
z = a + b i, a general comple number

(a,b), alternative form of a complex number z = a – b i, conjgate of z
|z| = modulus of complex number, Arg(z)= principle argument of z
z = r (cos  + i sin ), polar form of z, z = r ei , alternative polar form of z.