BASIC MATHEMATICALSession 2

Course : S0912 - Introduction to Finite Element MethodYear : 2010

Bina Nusantara

COURSE 2

Content:• Matrix• Vector Space• Basic Tensor

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MATRIX OPERATIONBASIC 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*aijzij = (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

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MATRIX OPERATIONBASIC OPERATION (REMINDER)• Determinant:

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MATRIX OPERATIONBASIC OPERATION (REMINDER)• Determinant:

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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 .

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MATRIX OPERATION

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MATRIX OPERATION

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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

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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.

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VECTOR SPACESeveral 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 thatX+(-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.

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VECTOR SPACE

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BASIC TENSOR

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BASIC TENSOR

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BASIC TENSOR

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BASIC TENSOR