Question

Let \(F \subseteq E\) be fields, \(n \in N, V=F^{n}, b \in V\), and \(A \in F^{n \times n}\). (i) Let \(a=\left(A^{i} b\right)_{i \in N} \in V^{N}\). Prove that the recursion order of \(a\) is the smallest \(r \in N\) such that \(b, A b, \ldots, A^{\prime} b\) are linearly dependent in \(F^{n}\). (ii) Let \(r \leq n\), and prove that vectors \(b_{0}, \ldots, b_{r-1} \in F^{n}\) are linearly dependent in \(F^{n}\) if and only if they are linearly dependent in \(E^{n}\). Hint: Gaussian elimination. Conclude that the minimal polynomial of \(a\) over \(F\) is the same as over \(E\).

Short answer

Recursion order is the smallest \(r\) such that \(b, Ab, \ldots, A^r b\) are dependent in \(F^n\). Linear dependence over \(F\) implies the same dependence over \(E\), ensuring identical minimal polynomials.

Step-by-step solution

Understanding Recursion Order

The recursion order of the sequence \(a = \left(A^i b\right)_{i \in \mathbb{N}}\) is the smallest integer \(r\) for which the vectors \(b, Ab, \ldots, A^{r}b\) are linearly dependent in \(F^n\). This means there exist coefficients \(c_0, c_1, \ldots, c_r\) not all zero such that \(c_0 b + c_1 Ab + \cdots + c_r A^r b = 0\). The task requires determining the smallest such \(r\).

Proving Linear Dependence over F

For part (i), note that if \(b, Ab, \ldots, A^r b\) are linearly dependent in \(F^n\), then there must be a non-trivial linear combination of these vectors that sums to zero. This implies that the sequence \(a\) satisfies a recurrence relation of order \(r\) with initial conditions derived from these vectors being linearly dependent.

Gaussian Elimination Strategy

For part (ii), consider the vectors \(b_0, \ldots, b_{r-1}\) being linearly dependent in \(F^n\). By Gaussian elimination, we can change the basis and determine if a linear combination of these vectors leads to the zero vector. This is facilitated by expressing the system of linear equations as a matrix and then performing row operations.

Extending to Field E

In the finite-dimensional space, linear dependence is preserved when extending the scalars from field \(F\) to field \(E\). This means if \(b_0, \ldots, b_{r-1}\) are linearly dependent in \(F^n\), they remain dependent in \(E^n\) because the operations involved (like row reductions) are preserved.

Minimal Polynomial Equivalence

The minimal polynomial of sequence \(a\) over field \(F\) is the polynomial of least degree which has \(a\) as a root sequence. The same steps show that when moving to field \(E\), the minimal polynomial does not change. This is because the dependence relation and hence the characteristic polynomial (which influences the minimal polynomial) remain the same.

Key concepts

Recursion Order

In a sequence, the recursion order is the smallest number that describes a pattern based on previous terms. Imagine you have a sequence created by applying a matrix transformation many times. The recursion order here is the smallest integer, say \( r \), for which you find that \( b, A b, ..., A^r b \) are linearly dependent. This means they can be expressed in a linear combination that sums to zero.
In simpler terms, when we talk about linear dependence, it means one of these vectors can be written as a combination of the others. Recursive sequences are quite logical in nature once you determine this pattern or order, as it helps in predicting subsequent terms from the preceding ones.
Understanding recursion order is key for identifying the nature of a sequence’s growth, as related to linear transformations.

Gaussian Elimination

Gaussian elimination is a technique used to solve systems of linear equations. It involves using row operations to transform a matrix into a simpler form, making it easier to find solutions.
In our exercise, Gaussian elimination helps show that linear dependence in vectors \( b_0, ..., b_{r-1} \) remains the same whether considered over a smaller field \( F \) or an extended field \( E \).

• The process begins by constructing a matrix from the given vectors.
• Then, systematically apply row operations to obtain the row-echelon form.
• In this form, you can observe whether a row of zeros exists, indicating linear dependence.

By maintaining these operations, the dependability checks remain consistent even when the field expands from \( F \) to \( E \). This fundamental property aids in understanding broader connections in linear algebra.

Minimal Polynomial

A minimal polynomial is the polynomial of the lowest degree that a vector satisfies when using a linear transformation. For a sequence \( a \) generated by applying transformations like \( A^i b \), the minimal polynomial helps relate \( a \) to the polynomial characteristics of a field.
When extending the field from \( F \) to \( E \), the minimal polynomial of \( a \) doesn't change. This is because the zero relations created by this polynomial in \( F \) remain valid in \( E \). Minimal polynomials are handy because they summarize crucial information about linear transformations: they help find eigenvalues and determine matrix structures.
It's worth noting that a minimal polynomial not only tells us about the recursive nature of sequences but also encompasses the linear dependency features within the vectors. They provide insight into the essence of transformations within finite dimensions, bridging basic algebraic principles with complex field extensions.