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 fi ne four fundamental spaces associated with matrix A Definition 22 . ( Column Space): The space spanned by column vectors of matrix A is de fi 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 y = ¯ 0 is called as null space of matrix A and denoted as N ( A T ) .
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