By Gilbert Strang

This informally written textual content presents scholars with a transparent advent into the topic of linear algebra. themes lined contain matrix multiplication, row aid, matrix inverse, orthogonality and computation. The self-teaching publication is loaded with examples and photographs and gives a wide range of probing difficulties, accompanying options, and a thesaurus.

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LINEAR ALGEBRA DUALS AND ADJOINTS Let V be a vector space over F. 1: V* = HomF(V, F) is called the dual of V. 25 that V* is finite dimensional with dim V* = dim V. 2. 2: Let V be finite dimensional. Then dim V* = dim V. Proof Let ocj be a basis of V. , n, define an = P is the isomorphism determined element xr e V* by czr = by the basis and P F is the natural projection onto the ith coordinate of P. Thus, if x = + + Here V —. then xr is given by cxt(x1cz1 + ... + = x1 = 0. Let is a basis of V*. Suppose We claim that z* = .

20: F" a)' is a commutative diagram of isomorphisms. 20 is commutative. 15. We need only argue M(b, a) is an -÷ F" is an isomorphism. invertible matrix. Then the map M(b, a)t = SM(La): Now change of basis matrices M(ö, a) are always invertible. 18. For any fJeV, we have M(a, b)M(ö, a)[fl]5 = M(a, b)[fl]a = [fl]5. This equation easily implies M(a, ö)M(ö, a) = I,,, the n x n identity matrix. 6. 21: In this example, we construct a vector space isomorphic to V". Let V be a vector space over F, and let n e N.

16 commutes. Only the uniqueness of I remains to be proved. If T' e Hom(V/W, V') is another map for which T'H = T, then I = T' on Im H. But H is surjective. Therefore, I = T'. 17: Suppose T e V'). Then Tm T V/ker T. Proof We can view T as a surjective, linear transformation from V to Tm T. 18, H is the natural map from V to V/ker T. We claim I is an isomorphism. Since IH = T and T: V —+ Tm T is surjective, I is surjective. Suppose & e ker I. Then T(cz) = TH(cz) = 1(ä) = 0. Thus, e ker T. But, then fl(cz) = 0.