nutshell

# nutshell - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Linear Algebra In A Nutshell 685 LINEAR ALGEBRA IN A NUTSHELL One question always comes on the ﬁrst day of class. Do I have to know linear algebra ?” My reply gets shorter every year: You soon will .” This section brings together many important points in the theory. It serves as a quick primer, not an oﬃcial part of the applied mathematics course (like Chapter 1 and 2). This summary begins with two lists that use most of the key words of linear algebra. The ﬁrst list applies to invertible matrices. That property is described in 14 diﬀerent ways. The second list shows the contrast, when A is singular (not invertible). There are more ways to test invertibility of an n by n matrix than I expected. 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 Ax = 0 has one solution x =0 Ax = 0 has inﬁnitely many solutions Ax = 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 A has rank r<n The reduced row echelon form is R = IR ha s at least one zero row The column space is all of R n The column space has dimension r<n The row space is all of R n The row space has dimension r<n All eigenvalues are nonzero Zero is an eigenvalue of A A T A is symmetric positive deﬁnite A T A is only semideﬁnite A has n (positive) singular values A has r<n singular values Now we take a deeper look at linear equations, without proving every statement we make. The goal is to discover what Ax = b really means. One reference is my textbook Introduction to Linear Algebra , published by Wellesley-Cambridge Press. That book has a much more careful development with many examples (you could look at the course page, with videos of the lectures, on ocw.mit.edu or web.mit.edu/18.06 ). The key is to think of every multiplication Ax , a matrix A times a vector x ,asa combination of the columns of A : Matrix Multiplication by Columns ± 12 36 ²± C D ² = C ± 1 3 ² + D ± 2 6 ² = combination of columns . Multiplying by rows, the ﬁrst component C +2 D comes from 1 and 2 in the ﬁrst row of A . But I strongly recommend to think of Ax aco lumn at at ime .N o t
686 Linear Algebra In A Nutshell x =(1 , 0) and x =(0 , 1) will pick out single columns of A : ± ²± ² 12 1 12 0 =ﬁ r s tc o l um n ± ²± ² =

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nutshell - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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