Why two vectors' covariance is the dot product of these two vectors
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I am trying to understand the OLS property that SST(Total sum of squares) = SSE (explained sum of squares) + SSR (residual sum of squares).
One of the steps is to prove that sample covariance of residuals and the fitted values y_hat is zero. One of the post on stack exchange says that the covariance of residuals and y_hat is actually their dot product. I am surprised to learn this..
So I looked for the handouts online, and one of them says that the two vectors' covariance is the same as their inner product/dot product. But I don't see why and how we reached to this conclusion.
The link is here: https://mathcs.clarku.edu/~djoyce/ma217/covar.pdf
I am wondering if there is a simple way to understand why the inner product of two vectors is the covariance?
Thank you very much!
linear-algebra inner-product-space covariance
asked Jan 6 at 19:29
commentallez-vouscommentallez-vous
2149
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add a comment |
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I am trying to understand the OLS property that SST(Total sum of squares) = SSE (explained sum of squares) + SSR (residual sum of squares).
One of the steps is to prove that sample covariance of residuals and the fitted values y_hat is zero. One of the post on stack exchange says that the covariance of residuals and y_hat is actually their dot product. I am surprised to learn this..
So I looked for the handouts online, and one of them says that the two vectors' covariance is the same as their inner product/dot product. But I don't see why and how we reached to this conclusion.
The link is here: https://mathcs.clarku.edu/~djoyce/ma217/covar.pdf
I am wondering if there is a simple way to understand why the inner product of two vectors is the covariance?
Thank you very much!
linear-algebra inner-product-space covariance
asked Jan 6 at 19:29
commentallez-vouscommentallez-vous
2149
$endgroup$
1
$begingroup$
A inner product is defined by various axioms. The covariance satisfies those axioms and is therefore an inner product for the space.
$endgroup$
– Karl
Jan 18 at 20:23
add a comment |
$begingroup$
I am trying to understand the OLS property that SST(Total sum of squares) = SSE (explained sum of squares) + SSR (residual sum of squares).
One of the steps is to prove that sample covariance of residuals and the fitted values y_hat is zero. One of the post on stack exchange says that the covariance of residuals and y_hat is actually their dot product. I am surprised to learn this..
So I looked for the handouts online, and one of them says that the two vectors' covariance is the same as their inner product/dot product. But I don't see why and how we reached to this conclusion.
The link is here: https://mathcs.clarku.edu/~djoyce/ma217/covar.pdf
I am wondering if there is a simple way to understand why the inner product of two vectors is the covariance?
Thank you very much!
linear-algebra inner-product-space covariance
asked Jan 6 at 19:29
commentallez-vouscommentallez-vous
2149
$endgroup$
I am trying to understand the OLS property that SST(Total sum of squares) = SSE (explained sum of squares) + SSR (residual sum of squares).
One of the steps is to prove that sample covariance of residuals and the fitted values y_hat is zero. One of the post on stack exchange says that the covariance of residuals and y_hat is actually their dot product. I am surprised to learn this..
So I looked for the handouts online, and one of them says that the two vectors' covariance is the same as their inner product/dot product. But I don't see why and how we reached to this conclusion.
The link is here: https://mathcs.clarku.edu/~djoyce/ma217/covar.pdf
I am wondering if there is a simple way to understand why the inner product of two vectors is the covariance?
Thank you very much!
linear-algebra inner-product-space covariance
linear-algebra inner-product-space covariance
asked Jan 6 at 19:29
commentallez-vouscommentallez-vous
2149
asked Jan 6 at 19:29
commentallez-vouscommentallez-vous
2149
asked Jan 6 at 19:29
commentallez-vouscommentallez-vous
2149
asked Jan 6 at 19:29
commentallez-vouscommentallez-vous
2149
asked Jan 6 at 19:29
commentallez-vouscommentallez-vous
2149
2149
1
$begingroup$
A inner product is defined by various axioms. The covariance satisfies those axioms and is therefore an inner product for the space.
$endgroup$
– Karl
Jan 18 at 20:23
add a comment |
1
$begingroup$
A inner product is defined by various axioms. The covariance satisfies those axioms and is therefore an inner product for the space.
$endgroup$
– Karl
Jan 18 at 20:23
1
1
$begingroup$
A inner product is defined by various axioms. The covariance satisfies those axioms and is therefore an inner product for the space.
$endgroup$
– Karl
Jan 18 at 20:23
$begingroup$
A inner product is defined by various axioms. The covariance satisfies those axioms and is therefore an inner product for the space.
$endgroup$
– Karl
Jan 18 at 20:23
add a comment |
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$begingroup$
A inner product is defined by various axioms. The covariance satisfies those axioms and is therefore an inner product for the space.
$endgroup$
– Karl
Jan 18 at 20:23