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[M] Show that the columns of the matrix $A$ are orthogonal by making an appropriate matrix calculation. State the calculation you use.$A=\left[\begin{array}{rrrr}{-6} & {-3} & {6} & {1} \\ {-1} & {2} & {1} & {-6} \\ {3} & {6} & {3} & {-2} \\ {6} & {-3} & {6} & {-1} \\ {2} & {-1} & {2} & {3} \\ {-3} & {6} & {3} & {2} \\ {-2} & {-1} & {2} & {-3} \\ {1} & {2} & {1} & {6}\end{array}\right]$

Thus, the columns of $A$ are orthogonal.

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 2

Orthogonal Sets

Vectors

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Okay, This question asks us to show that this matrix a has orthogonal columns. So to start, what I have here is I just put this matrix and python here and just to show, I'm just going to print out a real quick and as we can see our matrix A shows up here just like we expect. And then we want to quickly check that this matrix is orthogonal in its columns at least. And we can do this using the fact that adopt product of any two different columns should give a zero, and we can check that by quickly doing a transpose times a So what we're going to dio is we're gonna prince, product of multiplying two matrices of a transpose. I'm a And if we do that, we get zeros everywhere except on the diagonal. And what this means is that the dot product of any two unlike columns, gives us zero and the ones on the diagonal just give us the magnitude of each column vector. So therefore, we have orthogonal columns. Since our matrix right here looks like the identity

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