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Alright friends, linear algebra question:
x is a vector, y is a vector, A is a matrix, (dot) is dot product
If x (dot) (Ay) = 0 for all x and y, then dim(im(A)) = ?
Friends tell me 0, I assumed zero, but my reasoning is flawed and I'm really not sure what implies that A would be a zero matrix
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>>273640
its probably 0 dude
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If x is in the image of A, then, by definition, there exists a vector y such that x = Ay.
But then
x dot Ay = 0
implies
x dot x = 0
|x|^2 = 0
|x| = 0
x = 0
So the image of A is equal to {0}.
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>>273640
Just take x and y as different elementary vectors(the ones that have only a single one and the rest are zeroes). If x equals 1 at ith entry and the rest entries are zero, and y equals 1 at jth entry and rest of the entries are zeroes, then you get the ijth element of A upon x(dot)Ay. Hence for all aij in A, aij equals 0. So A is a zero matrix.
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