By Jorge Rebaza
ISBN10: 1118229622
ISBN13: 9781118229620

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Additional resources for A First Course in Applied Mathematics
Sample text
51 The correspondence between a linear transformation and a matrix is true not only for the spaces W1 and R m , but for any vector spaces V of dimension n and W of dimension m. Since there is a correspondence between linear transformations and matrices, then the same is true for some definitions in the two cases. Given a linear transformation T : V —> W, we say that the vector space V is the domain and the vector space W is the codomain of T. The set R(T) = {yeW: y = T(x)} is called the range or image of T.
B) If B is nonsingular, we have rank(A) = mnkiABB'1) < rank(AjB). 38 BASICS OF LINEAR ALGEBRA By combining this with the first inequality in (a), we conclude that ra,nk(AB) = rank(A). The proof of the last statement is similar. □ We now define the null space of A, denoted as N(A). 45 The null space of a matrix Am x n is defined as the vector subspace of R n : N(A) = {x e Rn : Ax = 0 } . 57 1 2 From the definition, we see that the null space is the set 2 4 3 6 {[xi X2 xs]T : x\ + 2x2 + 3^3 = 0}.
1 implies the following important result. 48 (Dimension Formula). Consider anm x n matrix A. Then, dimcol(A) + dimAT(^l) = n. 58 A = ri 4 7i 2 5 8 . Observe that we can write the third column a^ as a combi_3 6 9J nation of the first two oi, 0,2 as as = 2a,2 — a\. And since these first two columns are linearly independent, this implies that co\(A) is 2dimensional. Geometrically then, col(^4) is the plane — x + 2y — z = 0 passing through the origin, with normal vector [—1 2 — 1] T . 48, the null space N(A) is onedimensional.
A First Course in Applied Mathematics by Jorge Rebaza
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