Kernel of GL(n,F) acting on F^n

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In summary, the question is about the induced action of GL(n,F) on the set of k-dimensional subspaces of F^n and whether the kernel of this action consists of scalar matrices. The answer is that if g in GL(n,F) is not a scalar matrix, then a k-dimensional subspace can always be constructed that is not fixed by g. This is done by extending a pair of vectors, one of which is not proportional to the other, to a basis and constructing subspaces from the basis vectors.
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Homework Statement



Suppose GL(n,F) acts on F^n in the usual way. Consider the induced action on the set of all k-dimensional subspaces of F^n. What's the kernel of this action? Is it faithful

The Attempt at a Solution



Well, I anticipate that the kernel of this action consists of scalar matrices, that is scalar multiples of the identity matrix. The question is how to prove that if g in GL(n,F) is a not a scalar matrix then we can always construct a k-dimensional subspace not fixed by g.
 
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If A is not a multiple of the identity then there is a f such that Af is not proportional to f. Now, assuming k<n, you can construct a k-dimensional subspace such that f is in this subspace and Af is in a complementary subspace. For instance you extend the pair f,Af to a basis and construct your subspaces out of the basis vectors.
 

FAQ: Kernel of GL(n,F) acting on F^n

What is the Kernel of GL(n,F) acting on F^n?

The Kernel of GL(n,F) acting on F^n refers to the set of matrices in GL(n,F) that act as the identity transformation on F^n. In other words, the Kernel consists of all matrices that do not change the vector space F^n when multiplied with any vector in F^n.

How is the Kernel of GL(n,F) acting on F^n related to linear transformations?

The Kernel of GL(n,F) acting on F^n is closely related to linear transformations, as it represents the set of matrices that do not produce any change in the vector space F^n when acting as a linear transformation. This means that the Kernel is a crucial concept in understanding the behavior of linear transformations.

What is the significance of the Kernel of GL(n,F) acting on F^n in mathematics?

The Kernel of GL(n,F) acting on F^n plays a crucial role in various mathematical fields, including linear algebra, abstract algebra, and group theory. It is used to study the structure and properties of linear transformations and their associated vector spaces, making it a fundamental concept in these fields.

Can the Kernel of GL(n,F) acting on F^n be empty?

Yes, the Kernel of GL(n,F) acting on F^n can be empty, but only in certain cases. For example, if the vector space F^n is trivial, then the Kernel will be empty. However, in most cases, the Kernel will not be empty, as there will always be at least one matrix that acts as the identity transformation on F^n.

How can the Kernel of GL(n,F) acting on F^n be computed?

The Kernel of GL(n,F) acting on F^n can be computed by finding the null space of the transformation matrix. This can be done by solving a system of linear equations or by using various computational tools such as Gaussian elimination or matrix operations. Alternatively, the Kernel can also be determined by finding the basis vectors of the null space.

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