Linear Algebra Review Sheet 1 [19/10/2015]
Equations
Vector Spaces
Transpose: (A + B)T = B T + AT
Inverse: (AT )1 = (A1 )T
Orthogonal: A1 = AT
Symmetric Matrix: A = AT
Diagonal Matrix:
Scalar Matrix:
Triangular Matrix:
Eigen Vector & Eigen Value = ???
Axioms for Vector Addition():
1. x & y in V, x+y in V
2. x + y = y + x, commutative
3. x + (y + z) = (x + y) + z, associative
4. 0 + x = x + 0 = x, zero vector
5. x + (x) = (x) + x = 0, Negative vector
Axioms for Scalar Multiplication:
1. k & x in V, kx in V
2. k(x + y) = kx + ky, distributive
3. (k1 + k2 )x = k1 (x) + k2 (x), distributive
4. k1 (k2 .x) = (k1 .k2 )x
5. 1.x = x
Criterion for Subspace:
1. x & y in W, x+y in W
2. x in W, kx in W
Dot Product
Dot Product: a.b = a1 .b1 + a2 .b2 + a3 .b3 = |a||b|cos
a1
Directional Cosines: cos = |a|
b
Component of a on b: a. |b|
Projection of a on b:
b
)
(compb a)( |b|
( a.b
)b
b.b
Properties of Dot Product
a.b = 0, a = 0 or b = 0
a.b = b.a
a.a 0
a.a = |a|2
Cross Product
a b = 0, if a = 0 or b = 0
a b = |a||b| sin
a b = (b a)
a b = 0, a & b are parallel
|a b| : Area of Parallelogram
1
|a b| = P1 P2 P2 P3 : Area of Triangle
2
|a.(b c)| : Volume of Trapezoid
a.(b c): Scalar Triple Product
a.(b c) = 0: Coplanar
a (b c) = (a.c)b (a.b)c: Vector Triple Product
Lines and Space in 3-Space
Equation of Line:
Vector Equation
r = r2 + t.a or
r r2 = t.a,t = parameter,a = direction vector
Parametric Equation
< x, y, z >=< x2 + t(x2 x1 ), y2 + t(y2 y1 ), z2 +
t(z2 z1 ) >=< x2 + a1 t, y2 + a2 t, z2 + a3 t >
Symmetric Equation t =
xx2
a1
yy2
a2
zz2
a3
Cartesian Equation:
Equation of Plane: a(x x1 ) + b(y y1 ) + c(z z1 ) = 0
Plane with Normal Vector:
Vector Normal to Plane(3x 4y + 10z 8 = 0) =
n = 3i 4j + 10k
Three Point Determine Plane
Normal vector: n = ai + bj + ck
Three Collinear Points: [(r2 r1 ) (r3 r1 )].(r r1 ) = 0
Theorem: ax + by + cz + d = 0 is a plane with normal vector
n = ai + bj + ck
Three Points (Plane)
Non-Trivial:Number of Equations is less then Number
of Variables i.e. infinite many solutions
Infinite Many > Not-Unique > Matrix Singular
Two Properties of Homogeneous System
X1 is a solution of AX = 0, then cX1 too
X1 & X2 are solutions of AX = 0, then X1 & X2 too
Rank of a Matrix
Rank is the maximum number of linearly independent
row vectors.
Max No of Independent Cols = Max No of Independent
Rows
Consistency of AX = B
Rank A = Rank A|B
Properties of Determinant
Determinant of Transpose, det(AT ) = det(A)
Two Identical Rows, det(A) = 0
Vector Calculus
Zero Row or Cols, det(A) = 0
Chain Rule: z = f (u, v), u = g(x, y), v = h(x, y)
z u
z v z
z u
z v
z
= u
+ v
,
= u
+ v
x
x
x y
y
y
Interchange Rows or Cols, det(B) = det(A)
Constant Multiple of Row, det(B) = k.det(A)
Gradient: F (x, y, z) = F
i + F
j + F
k
x
y
z
Directional Derivative:
Du f (x, y) = f (x, y).u, u = cos i + sin j(unit vector)
Divergence: div(F ) = P
+ Q
+ R
, F = P i + Qj + Rk
x
y
z
( R
y
( P
z
R
)j
x
( Q
x
i
j
k
curl(F ) = F = x
y
z .
P
Q
R
H
RR Q
Greens Theorem: c P dx + Qdy = R ( x P
)dA
y
Curl: curl (F ) =
Q
)i
z
Systems of Linear Equations
Types of Equations:
1. Homogeneous
2. Non-Homogeneous
Elimination Methods:
1. Gauss Elimination
2. Gauss Jordan Elimination
(Row Reduced Echelon Form)
Solution of Linear Equation:
Non-Homogeneous
Consistent
Inconsistent
Homogeneous
Always Consistent
(Trivial or Non-Trivial)
Trivial > Unique Solution > Matrix
Non-Singular
P
y
Determinant of Matrix Product,
det(AB) = det(A).det(B)
Determinant is Unchanged, row operations
)k
Determinant of Triangular Matrix,
det(A) = a11 .a22 .a33 ...ann
Property of Cofactor,
ai1 Ck1 + ai2 Ck2 + ... + ain Ckn = 0
Inverse of Matrix
When Inverse exist (Matrix non-singular): Solution is
Unique
Eigen Value Problems
Complex Eigen Values: Vectors Conjugate of Each other
A (Singular): Zero Eigen Value
A : & x, A1 : 1 & x
Eigen Value (Upper, Lower, Diagonal): Diagonal
Entries
Orthogonal Matrices
A(symmetric+real values): Eigen values Real
A(symmetric): Eigen vectors corresponding to different
eigen value are orhtogonal
A(orthogonal): Column forms orthonormal set
Diagonalization
c 2015 RiZ
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