By Bacco M., Mocellin V.

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**Example text**

Finally, the determinant of the upper triangular matrix equals the product of the diagonal entries. 2: Basic Properties of Determinants 1 4 2 ■2 4 7 6 1 2 = 1 4 2 0 12 0 0 11 133 I = 133. IT Proceeding in a similar fashion in (b) and (c), we find the answers to be 132 and -26 2. respectively. If one row (or column) of a determinant is a scalar multiple of another row (or column), show that the determinant is zero. Solution. Suppose that row operation i equals R- - cR- , * J c times row j . Then if we apply the row it will not change the value of the determinant, but the i th row will then consist of O's.

D = 0 (z - t/)(y - z)(z - x)(x + y + z). x , y , z , 8. Let a = b. = 1. n denote the "bordered" n % n determinant Prove that 0 a 0 0 .. 0 0 0 b 0 a 0 .. 0 0 0 0 b 0 a 0 0 0 0 0 0 0 . b 0 a 0 0 0 0 .. 0 b 0 £>2 x = 0 and D2n = (-ab)n. Solution. Expand by row 1, and then expand the resulting determinant by column 1 to get 2) = (-a)bDn 0 , that is, £> = (-aft)D can use the recurrence relation to show that D 2 = -aft , so that D2 = (-a6)n. 0. ^on-l Now = ZX. = 0, so we ° for all n . 2: Basic Properties of Determinants 9.

Chapter Four: Introduction to Vector spaces 56 6. Prove that the vector spaces C [0, 1] and P (F ) are infinitely generated where F is any field. Solution. Suppose that C [0, 1] can be generated by / , , . , / „ . 4, that some linear combination of these be 0, which is certainly wrong. i = 0, lb . ) The same proof works for P (F ). o 7. Let A and B be vectors in R .. Show that A and B generate 2 R if and only if neither is a scalar multiple of the other. Interpret this result geometrically. Solution.