Show that there is a basis C of V so that C* = Λ

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In summary: I don't think there is anything to improve. (Smile)In summary, the conversation discusses the transformation matrices of bases in a $n$-dimensional $K$-vector space, and shows that the transformation matrix from $B^{\star}$ to $C^{\star}$ is equal to $\left (S^{-1}\right )^T$. Furthermore, it is shown that the matrix for the transformation of a basis must exist.
  • #1
mathmari
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Hey! :eek:

Let $K$ be a field and $V$ a $n$-dimensional $K$-vector space with basis $B=\{b_1, \ldots , b_n\}$. $V^{\star}$ is the dual space of $V$. $B^{\star}$ is the dual basis corresponding to $B$ of $V^{\star}$.

  1. Let $C=\{c_1, \ldots , c_n\}$ be an other basis of $V$ and $C^{\star}$ its dual basis. Let $S=(s_{ij})_{1\leq i,j\leq n}$ be the transformation matrix of basis from $B$ to $C$, i.e. $\displaystyle{c_j=\sum_{i=1}^ns_{ij}b_i}$ for $j\in \{1, \ldots , n\}$.
    Show that $(S^{-1})^T=:A=(a_{ij})_{1\leq i,j\leq n}$ is the transformation matrix of basis from $B^{\star}$ to $C^{\star}$ i.e. that $\displaystyle{c_j^{\star}=\sum_{i=1}^na_{ij}b_i^{\star}}$ for $j\in \{1, \ldots , n\}$.
  2. Let $\Lambda=\{\lambda_1, \ldots , \lambda_n\}$ a basis of the dual space $V^{\star}$.
    Show that there is a basis $C$ of $V$ so that $C^{\star}=\Lambda$.
I have done the following:
  1. We have that $A$ is the transformation matrix of basis from $B^{\star}$ to $C^{\star}$, i.e., that $\displaystyle{c_j^{\star}=\sum_{i=1}^na_{ij}b_i^{\star}}$ and we want to show that the transformation matrix is equal to $A:=\left (S^{-1}\right )^T$, right? (Wondering)

    We have the following:
    \begin{equation*}c_j^{\star}(c_k)=c_j^{\star}\left (\sum_{i=1}^{n}s_{ik}b_i\right )\ \overset{ c_j^{\star} \text{ linearform }}{ = } \ \sum_{i=1}^{n}s_{ik}c_j^{\star}\left (b_i\right )=\sum_{i=1}^{n}s_{ik}\sum_{\lambda=1}^na_{\lambda j}b_{\lambda}^{\star}(b_i)=\sum_{i=1}^{n}s_{ik}\sum_{\lambda=1}^na_{\lambda j}\delta_{\lambda i}=\sum_{i=1}^{n}s_{ik}a_{i j}\end{equation*}

    From that we get $\displaystyle{\sum_{i=1}^{n}s_{ik}a_{i j}=\delta_{jk}}$, since $c_j^{\star}(c_k)=\delta_{jk}$.

    From $\displaystyle{\sum_{i=1}^{n}a_{i j}s_{ik}=\delta_{jk}}$ it follows that $\displaystyle{\sum_{i=1}^{n}a_{ji}^Ts_{ik}=\delta_{jk}}$.

    Since $\delta_{jk}=\begin{cases}1 & \text{ if } i=k \\0 & \text{ otherwise } \end{cases}$ it follows that we get the identity matrix.

    So we get $A^T\cdot S=I_n$.

    Therefore we have that \begin{equation*}A^T=S^{-1}\Rightarrow \left (A^T\right )^T=\left (S^{-1}\right )^T\Rightarrow A=\left (S^{-1}\right )^T\end{equation*}


    Is everything correct? (Wondering)
  2. Could you give me a hint for that? (Wondering)

    Do we maybe do the following?

    Let $M$ the transformation matrix of basis from $B^{\star}$ to $\Lambda$.
    Then do we get from the first question that $M=\left (S^{-1}\right )^T$ ? Or does this only hold for the specific basis $C^{\star}$ ?

    (Wondering)
 
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  • #2
mathmari said:
I have done the following:
  1. We have that $A$ is the transformation matrix of basis from $B^{\star}$ to $C^{\star}$, i.e., that $\displaystyle{c_j^{\star}=\sum_{i=1}^na_{ij}b_i^{\star}}$ and we want to show that the transformation matrix is equal to $A:=\left (S^{-1}\right )^T$, right? (Wondering)

    We have the following:
    \begin{equation*}c_j^{\star}(c_k)=c_j^{\star}\left (\sum_{i=1}^{n}s_{ik}b_i\right )\ \overset{ c_j^{\star} \text{ linearform }}{ = } \ \sum_{i=1}^{n}s_{ik}c_j^{\star}\left (b_i\right )=\sum_{i=1}^{n}s_{ik}\sum_{\lambda=1}^na_{\lambda j}b_{\lambda}^{\star}(b_i)=\sum_{i=1}^{n}s_{ik}\sum_{\lambda=1}^na_{\lambda j}\delta_{\lambda i}=\sum_{i=1}^{n}s_{ik}a_{i j}\end{equation*}

    From that we get $\displaystyle{\sum_{i=1}^{n}s_{ik}a_{i j}=\delta_{jk}}$, since $c_j^{\star}(c_k)=\delta_{jk}$.

    From $\displaystyle{\sum_{i=1}^{n}a_{i j}s_{ik}=\delta_{jk}}$ it follows that $\displaystyle{\sum_{i=1}^{n}a_{ji}^Ts_{ik}=\delta_{jk}}$.

    Since $\delta_{jk}=\begin{cases}1 & \text{ if } i=k \\0 & \text{ otherwise } \end{cases}$ it follows that we get the identity matrix.

    So we get $A^T\cdot S=I_n$.

    Therefore we have that \begin{equation*}A^T=S^{-1}\Rightarrow \left (A^T\right )^T=\left (S^{-1}\right )^T\Rightarrow A=\left (S^{-1}\right )^T\end{equation*}

Hey mathmari!

It seems correct to me. (Nod)

mathmari said:
2. Could you give me a hint for that? (Wondering)

Do we maybe do the following?

Let $M$ the transformation matrix of basis from $B^{\star}$ to $\Lambda$.
Then do we get from the first question that $M=\left (S^{-1}\right )^T$ ? Or does this only hold for the specific basis $C^{\star}$ ?

Isn't it that we can rather construct $S = (M^{-1})^T$ by the reverse argument of the first question? (Wondering)

$M$ must exist since $V^*$ is a vector space.
And with the constructed $S$ we can find a basis $C$ in $V$ that is the dual of $\Lambda$. (Thinking)
 
  • #3
I like Serena said:
Isn't it that we can rather construct $S = (M^{-1})^T$ by the reverse argument of the first question? (Wondering)

$M$ must exist since $V^*$ is a vector space.
And with the constructed $S$ we can find a basis $C$ in $V$ that is the dual of $\Lambda$. (Thinking)

Why must $M$ exist? Could you explain it further to me? (Wondering)

So $M$ is the transformation matrix of basis from $B^{\star}$ to $\Lambda$.

Let $S:=(M^{-1})^T$.

Let $C$ be the basis that we get by the transformation matrix $S$ from the basis $B$.

From the first question we have then that $\left (S^{-1}\right )^T$ is the transformation matrix of basis from the dual basis $B^{\star}$ to the dual basis $C^{\star}$. Since $\left (S^{-1}\right )^T=M$ and $M$ is the transformation matrix from $B^{\star}$ to $\Lambda$, it follows that $\Lambda=C^{\star}$. Is everything correct? Could we improve something? (Wondering)
 
  • #4
mathmari said:
Why must $M$ exist? Could you explain it further to me?

Lemma: The matrix for the transformation of a basis B to a basis C in a vector space must exist.
Proof: Observe that every basis vector in C must have a unique linear representation of basis vectors in B.
When we combine those linear representations into a matrix we have the matrix for the basis transformation. (Nerd)

mathmari said:
So $M$ is the transformation matrix of basis from $B^{\star}$ to $\Lambda$.

Let $S:=(M^{-1})^T$.

Let $C$ be the basis that we get by the transformation matrix $S$ from the basis $B$.

From the first question we have then that $\left (S^{-1}\right )^T$ is the transformation matrix of basis from the dual basis $B^{\star}$ to the dual basis $C^{\star}$. Since $\left (S^{-1}\right )^T=M$ and $M$ is the transformation matrix from $B^{\star}$ to $\Lambda$, it follows that $\Lambda=C^{\star}$.

Is everything correct? Could we improve something?

It looks fine to me. (Nod)
 

1. How do you show that there is a basis C of V so that C* = Λ?

To show that there is a basis C of V so that C* = Λ, you need to prove that C* is equal to the dual space of V. This means that every linear functional on V can be written as a linear combination of elements in C*, and every element in C* can be uniquely represented as a linear combination of elements in V.

2. What is the significance of having a basis C of V where C* = Λ?

The significance of having a basis C of V where C* = Λ is that it allows for easier computations and proofs involving linear transformations and dual spaces. It also simplifies the process of finding the transpose of a matrix, as C* can be thought of as the rows of the transpose of the matrix whose columns are the basis vectors of C.

3. Can every vector space V have a basis C where C* = Λ?

Not all vector spaces V have a basis C where C* = Λ. This is because the condition C* = Λ requires that the vector space V is finite-dimensional and has a dual space. In infinite-dimensional vector spaces, such as function spaces, the concept of a dual space may not exist.

4. How can you find a basis C of V where C* = Λ?

To find a basis C of V where C* = Λ, you can use the Gram-Schmidt process. This process takes a set of linearly independent vectors and produces an orthogonal set with the same span. By applying this process to the standard basis of V, you can obtain a basis C where C* = Λ.

5. What are the applications of having a basis C of V where C* = Λ?

Having a basis C of V where C* = Λ has various applications in linear algebra, functional analysis, and other areas of mathematics and science. It can be used to simplify calculations involving linear transformations, find the transpose of a matrix, and prove theorems related to dual spaces. It also has applications in physics, engineering, and computer science, such as in signal processing and control systems.

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