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- Thread starter gsingh2011
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In summary, the conversation discusses the relationship between the determinant of a matrix and the cross product formula. The determinant is a unique, antisymmetric, and multilinear function on n vectors in n dimensions, and considering n-1 of the arguments (rows or columns) held fixed makes it a linear function on the remaining argument. This process identifies a pair of elements of the space with a third, defining a product operation that is antisymmetric and linear in both arguments and orthogonal to either of them, which are the defining properties of the cross product. In a finite dimensional real vector space with an inner product, any real linear function is of the form f(v)=u.v for some vector u, which explains the connection between the determinant and the cross

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It can be shown that for a finite dimensional real vector space with an inner product (like the dot product in R^3) any real linear function is of the form f(v)=u.v for some vector u. So if we consider the determinant of a 3x3 matrix as a function of its first column then this must be a dot product with some vector which depends on the other two columns. (By direct calculation it is the dot product with the vector whose components are 2x2 determinants, but you seemed to be asking for a more conceptual explanation). Since the determinant is linear in each column and antisymmetric (changes sign when any two rows or colums are exchanged) the vector whose dot product with the first column gives the determinant depends linearly on each of the other two columns and changes sign when they are reversed. Also since the determinant is zero when columns repeat, this combination of the last two columns is orthogonal to both of them (since the corresponding function of the first column is zero when it matches either of the other two columns). These conditions are enough to determine the cross product.

[More generally, in an n-dimensional space there is a generalization of the cross product defined not for pairs but for groups of n-1 vectors by taking as components the signed minor determinants which multiply by the entries of the first column to get the overall determinant of the nxn matrix with the given n-1 vectors as its remaining columns]

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Hi

gsingh2011 said:

I think

a determinant is usually a number, and is derived from a transformation matrix …

so

I can't think of a good answer to any of this, I can only think of more problems

… for example, the determinant of a product of two transformations is the product of the determinants, but what would the first "product" mean in this case?

… for example, the determinant of a product of two transformations is the product of the determinants, but what would the first "product" mean in this case?

Does anybody have a good answer to this?

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