Download vector space - Binus Repository

Course
Year
: S0912 - Introduction to Finite Element Method
: 2010
BASIC MATHEMATICAL
Session 2
COURSE 2
Content:
• Matrix
• Vector Space
• Basic Tensor
Bina Nusantara
MATRIX OPERATION
BASIC OPERATION (REMINDER)
• Addition:
Z = A + B; zij = aij + bij
• Substraction:
Z = A - B; zij = aij - bij
• Multiplication and division of a matrix by a scalar
zij = c*aij
zij = (1/c)*aij
• Multiplication:
Z = A*B, if # columns in A = # rows in B; zij = ai1* b1j + ai2* b2j + ai3* b3j + ... aim* bnj
• Transpose Operation
• Inverse Operation
Bina Nusantara
MATRIX OPERATION
BASIC OPERATION (REMINDER)
• Determinant:
Bina Nusantara
MATRIX OPERATION
BASIC OPERATION (REMINDER)
• Determinant:
Bina Nusantara
MATRIX OPERATION
Eigenvector & Eigenvalue:
Let A be a complex square matrix. Then if  is a complex
number and X a non–zero complex column vector
satisfying AX = X, we call X an eigenvector of A, while  is
called an eigenvalue of A. We also say that X is an
eigenvector corresponding to the eigenvalue .
Bina Nusantara
MATRIX OPERATION
Bina Nusantara
MATRIX OPERATION
Bina Nusantara
VECTOR SPACE
• A vector space is a mathematical structure formed by a collection of
vectors: objects that may be added together and multiplied ("scaled") by
numbers.
• Vector spaces are the subject of linear algebra and are well understood
from this point of view, since vector spaces are characterized by their
dimension
Bina Nusantara
VECTOR SPACE
• A vector space is a set that is closed under finite vector addition
and scalar multiplication. The basic example is -dimensional
Euclidean space , where every element is represented by a list of
real numbers, scalars are real numbers, addition is componentwise,
and scalar multiplication is multiplication on each term separately.
• For a general vector space, the scalars are members of a field , in
which case is called a vector space over .
• Euclidean -space is called a real vector space, and is called a
complex vector space.
Bina Nusantara
VECTOR SPACE
Several operation of vector space in order of X,Y,Z in V and any scalars r,s in F:
1. Commutativity:
X+Y=Y+X.
2. Associativity of vector addition:
(X+Y)+Z=X+(Y+Z).
3. Additive identity: For all X,
0+X=X+0=X.
4. Existence of additive inverse: For any X, there exists a -X such that
X+(-X)=0.
5. Associativity of scalar multiplication:
r(sX)=(rs)X.
6. Distributivity of scalar sums:
(r+s)X=rX+sX.
7. Distributivity of vector sums:
r(X+Y)=rX+rY.
8. Scalar multiplication identity:
1X=X.
Bina Nusantara
VECTOR SPACE
Bina Nusantara
BASIC TENSOR
Bina Nusantara
BASIC TENSOR
Bina Nusantara
BASIC TENSOR
Bina Nusantara
BASIC TENSOR
Bina Nusantara