Linear algebra determinant of linear operator

In summary, The problem is asking to prove that the determinant of a linear operator T can be defined as the determinant of its matrix representation with respect to any ordered basis. It also states that for any two ordered bases of V, the determinant of the matrix representation of T will be the same.
  • #1
pezola
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[SOLVED] linear algebra determinant of linear operator

Homework Statement



Let T be a linear operator on a finite-dimensional vector space V.
Define the determinant of T as: det(T)=det([T]β) where β is any ordered basis for V.

Prove that for any scalar λ and any ordered basis β for V that det(T - λIv) = det([T]β - λI).

Homework Equations



Another part of the problem yielded that for any two ordered bases of V, β and γ , that det([T]β) = det([T]γ).


The Attempt at a Solution



I need someone to help me understand the notation. I don't actually know what I am being asked to prove here.
 
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  • #2
T is the general linear transformation. [T] is the matrix that represents the linear transformation with respect to the basis B. If we change the basis, then we change the matrix since if TX = Y where X and Y are the coordinate vectors in K^n representing the abstract vectors x,y in V. (We can do with since any n dimensional vector space is isomorphic to K^n.) These coordinates are just the coefficients of the basis vectors.
ie:if x = a1v1 +a2v2 + .. + anvn where v1 ...vn is a basis then the coordinate vector for x just (a1,a2,a3,...,an). Obviously these coordinates change when the basis changes. The same holds for Y which means our matrix will need to change to reflect this.

Also, X and Y are column vectors so that the matrix multiplication is defined.
 

1. What is the definition of a linear algebra determinant?

A linear algebra determinant is a scalar value that can be calculated from a square matrix. It represents the magnitude by which the matrix scales the space it operates on.

2. What is a linear operator?

A linear operator is a mathematical function that operates on a vector space and satisfies the properties of linearity. It maps one vector to another vector in the same space.

3. How is the determinant of a linear operator calculated?

The determinant of a linear operator is calculated by finding the determinant of its corresponding matrix representation. This can be done using various methods, such as cofactor expansion or Gaussian elimination.

4. What is the significance of the determinant in linear algebra?

The determinant is an important concept in linear algebra as it provides information about the properties of a linear operator or matrix. It can indicate if a system of equations has a unique solution, and it is also used in finding eigenvalues and eigenvectors.

5. Can the determinant of a linear operator be negative?

Yes, the determinant of a linear operator can be negative. This indicates that the linear operator has a reflection component, meaning it flips the space it operates on along a certain axis.

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