# Linear Algebra: An Introduction with Concurrent Examples by A. G. Hamilton

By A. G. Hamilton

It is a stable, common advent to linear algebra. The "concurrent examples" procedure works good. My major difficulty is that the geometric elements of the topic are a bit slighted. Geometry is taken care of very in brief (e.g., regardless of the emphasis on labored examples, there's not a unmarried numerical instance of a selection of the space from a line to some degree or from a airplane to some degree) and fairly clumsily (e.g., utilizing the cosine formulation to end up a.b=|a||b|cos(theta), p. 108). This hampers the exposition later. for instance, Gram--Schmidt is brought purely in the direction of the top the place it's caused by means of the orthonormal eigenvector-matrix method of diagonalisation of symmetric matrices, hence belittling its geometric value. additionally, it kind of feels abnormal to have such an problematic dialogue of diagonalisation with no ever stating its use to find A^n.

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Sample text

EzEI =1. Hence B is the inverse of A. Our procedure for finding the inverse of A must therefore calculate for us the product EsEs _ I ... EzE I' This product can be written as EsEs-I ... EzEll, and this gives the hint. We convert A to 1 by certain elementary row operations. The same row operations convert 1 into A - I (if it exists). Explicitly, if I=Es ... EIA then A-1=Es ... Ell. 5 Find the inverse of the matrix [1 U. 0 1 2 1 0 2 1 0 0 0 1 2 0 1 0 1 2 0 0 0 1 1 0 2 1 0 0 0 1 0 2 1 0 0 1 (3)-(1) 0 0 1 0 0 2 -2 -1 I 1 0 0 1 0 0 2 1 0 2 -6 -2 -1 1 (3)-2 x (2) I 0 0 0 1 0 1.

I) [~ (ii) 2J 3 . [ ]. (vi) [I (x) (iii) 2 1 ~ 0 ~ (iv) (viii) [~~J [ 1 (v) 1] . (vii) 1 -2 -1 ] 4 . ]. 2 2 0 -1] -4 . 3 -~ U [ -1 -1 3 1 4]' 3 . 8 1 1 1 0 0 -2 -1 -1 -3 i] 2. Find the inverse of the diagonal matrix [ ~], ~~ o 0 c where a, band c are non-zero. 3. Let A, Band C be invertible matrices of the same size. Show that the inverse of the product ABC is C -1B -1 A-I. 4. Let x and y be p-vectors. Show that xyT is a p x p matrix and is singular. Pick some vectors x and y at random and verify that xyT is singular.

6. 7) that sums of symmetric matrices are symmetric, but products in general are not. There are three important rules about transposes. 36 Examples Let A and B be any p x p matrices. Then (AB)T = BTAT. To see this, let the (i,j)-entries of A and B be denoted by aij and bij respectively. 8 p L ajkbki. k=l The (i,j)-entry in BT AT is p L blal j, k= 1 where bl is the (i, k)-entry in BT and definition of the transpose, we have al j is the (k,j)-entry in AT. e. bkiajk, k=l ajkbki, k=l which is the same as the (i,j)-entry in (AB)T.