1. Matrices and Linear Systems Roughly speaking, matrix is a rectangle array We shall discuss existence and uniqueness of solution for a system of linear equation. The method of Gauss ellimination will be given to solve the system.

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30. Vector Spaces A quantity such as work, area or energy which is described in terms of magnitude alone is called a scalar. A quantity which has both magnitude and direction for its describtion is called a vector. A vector is an element of vector space.

31. Definiton: A vector space V in R is the set satisfying

32. Examples for vector spaces

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39. Dimension of a vector space V SpanS= All linear combinations of vectors of the subset S of V. A basis for V is a linearly independent subset S of V which spans the space V. That is, SpanS= V where S is lin. İndep. dimV= The number of vectors in any basis for V. V is finite-dimensional if V has a basis consisting of a finite number of vectors.

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48. Determinant Determinant is a function form square matrices to scalars. Our efficient computational procedure will be cofactor expansion.

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76. Examples

77. Linear Transformations Examples: Zero transform, identity operator, scalar-multiple operator,reflection, projection, rotation, differential transform, integral transform.

78. Representiation Matrix

79. Example

80. Example: Find the representiation matrix of

81. Range and Null (Kernel) spaces

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