# Further Properties of Abelian Integrals Attached to by W.V.D. Hodge By W.V.D. Hodge

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Extra info for Further Properties of Abelian Integrals Attached to Algebraic Varieties

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If you draw both versions, then you get a parallelogram with the sum of the vectors as the diagonal arrow who's tail starts at the origin. What is a parallelogram? you might ask, if your high school geometry is a bit murky. A parallelogram is a four sided figure with opposite sides parallel and equal in length. So, for example, the blue arrows representing the vector s are the same length and same direction. The green arrows representing the vector r have their same length and same direction. html [10/9/01 2:23:44 PM] More Practice u = ( -3, 2 )T, v = ( 1, -5 )T form the sum: w=u+v A good answer might be: w = ( -2, -3 )T More Practice Now draw u, v, and w on the graph paper, where, as before, u = ( -3, 2 )T, w = ( -2, -3 )T QUESTION 16: Continue to the next page when you have answered.

Type a number into one of the boxes, hit Enter, and see the results. html [10/9/01 2:23:32 PM] Commutative A good answer might be: ( 8, 4, 6 )T + ( 2, -2, 9 )T = ( 10, 2, 15 )T ( 2, -2, 9 )T + ( 8, 4, 6 )T = ( 10, 2, 15 )T Commutative You might suspect that the last problem is supposed to sneak in another math fact. You are right: Matrix addition is commutative. This means that a + b = b + a. This works for both row and column matrices of all dimensions. It is also true that: a + b + c = b + c + a = c + a + b = .....

You might ask, if your high school geometry is a bit murky. A parallelogram is a four sided figure with opposite sides parallel and equal in length. So, for example, the blue arrows representing the vector s are the same length and same direction. The green arrows representing the vector r have their same length and same direction. html [10/9/01 2:23:44 PM] More Practice u = ( -3, 2 )T, v = ( 1, -5 )T form the sum: w=u+v A good answer might be: w = ( -2, -3 )T More Practice Now draw u, v, and w on the graph paper, where, as before, u = ( -3, 2 )T, w = ( -2, -3 )T QUESTION 16: Continue to the next page when you have answered.