nutshell

nutshell - The column space is all of R n The column space...

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LINEAR ALGEBRA IN A NUTSHELL (( A is n by n )) Nonsingular Singular A is invertible A is not invertible The columns are independent The columns are dependent The rows are independent The rows are dependent The determinant is not zero The determinant is zero A x = 0 has one solution x = 0 A x = 0 has inﬁnitely many solutions A x = b has one solution x = A - 1 b A x = b has no solution or inﬁnitely many A has n (nonzero) pivots A has r < n pivots A has full rank r = n A has rank r < n The reduced row echelon form is R = I R has at least one zero row
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Unformatted text preview: The column space is all of R n The column space has dimension r &lt; n The row space is all of R n The row space has dimension r &lt; n All eigenvalues are nonzero Zero is an eigenvalue of A A T A is symmetric positive denite A T A is only semidenite A has n (positive) singular values A has r &lt; n singular values Each line of the singular column can be made quantitative using r . 568...
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