# An Introduction To Linear Algebra by Kenneth Kuttler

By Kenneth Kuttler

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However, which one is meant will be determined by the context in which they occur. These vectors have a significant property. 3 Let v ∈ Fn . Thus v is a list of numbers arranged vertically, v1 , · · · , vn . Then eTi v = vi . 22 makes perfect sense. It equals   v1  ..   .     (0, · · · , 1, · · · 0)   vi  = vi  .   ..  vn as claimed. 23. From the definition of matrix multiplication using the repeated index summation convention, and noting that (ej )k = δ kj     A1k (ej )k A1j    ..

24. 24 whenever one of the vectors is a scalar multiple of the other. It only remains to verify this is the only way equality can occur. 24 so it can be assumed both vectors are non zero. Then if equality is achieved, it follows f (t) has exactly one real zero because the discriminant vanishes. Therefore, for some value of t, a + tθb = 0 showing that a is a multiple of b. This proves the theorem. 23. This means that whenever something satisfies these properties, the Cauchy Schwartz inequality holds.

9. ♠ Prove that Im A = A where A is an m × n matrix. 10. ♠ Let A and be a real m × n matrix and let x ∈ Rn and y ∈ Rm . Show (Ax, y)Rm = x,AT y Rn where (·, ·)Rk denotes the dot product in Rk . T 11. ♠ Use the result of Problem 10 to verify directly that (AB) = B T AT without making any reference to subscripts. 12. Let x = (−1, −1, 1) and y = (0, 1, 2) . Find xT y and xyT if possible. 13. ♠ Give an example of matrices, A, B, C such that B = C, A = 0, and yet AB = AC.     1 1 1 1 −3 1 −1 −2 0  .