1. BASIC MATHEMATICAL Session 2 Course: S0912 - Introduction to Finite Element Method Year: 2010

2. Bina Nusantara COURSE 2 Content: Matrix Vector Space Basic Tensor

3. 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

4. Bina Nusantara MATRIX OPERATION BASIC OPERATION (REMINDER) Determinant:

5. Bina Nusantara MATRIX OPERATION BASIC OPERATION (REMINDER) Determinant:

6. 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.

7. Bina Nusantara MATRIX OPERATION

8. Bina Nusantara MATRIX OPERATION

9. 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

10. 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.

11. 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.

12. Bina Nusantara VECTOR SPACE

13. Bina Nusantara BASIC TENSOR

14. Bina Nusantara BASIC TENSOR

15. Bina Nusantara BASIC TENSOR

16. Bina Nusantara BASIC TENSOR