Chemical Engineering Hand Written_Notes_Part_28

Chemical Engineering Hand Written_Notes_Part_28 - 58 3....

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58 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES or equivalently the n column vectors are not linearly independent. Thus, both these geometric interpretations are consistent with each other. Now, to understand behavior of solutions of type (1.5), we can de f ne four fundamental spaces associated with matrix A Definition 22 . ( Column Space): Thespac espannedbyco lumnve c tor s of matrix A is de f ned as column space of the matrix and denoted as R ( A ) . Definition 23 . ( Row Space): The space spanned by row vectors of matrix A is called as row space of matrix A and denoted as R ( A T ) . Definition 24 . ( Null space): The set of all vectors x such that A x = ¯ 0 is called as null space of matrix A and denoted as N ( A ) . A non-zero null space is obtained only when columns of A are linearly de- pendent. If columns of A are linearly independent, then N ( A ) { ¯ 0 } . Definition 25 . ( Left Null Space) : The set of all vectors y such that A T
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Chemical Engineering Hand Written_Notes_Part_28 - 58 3....

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