Linear Algebra

Dual Vector Space and Dual Basis, Some Equality

Problem 282

Let $V$ be a finite dimensional vector space over a field $k$ and let $V^*=\Hom(V, k)$ be the dual vector space of $V$.
Let $\{v_i\}_{i=1}^n$ be a basis of $V$ and let $\{v^i\}_{i=1}^n$ be the dual basis of $V^*$. Then prove that
\[x=\sum_{i=1}^nv^i(x)v_i\] for any vector $x\in V$.

Add to solve later

Proof.

Recall that the dual basis $\{v^i\}_{i=1}^n$ consists of vectors $v^i \in V^*$ satisfying
\[v^j(v_i)=\delta_{i,j}, \tag{*}\] where $\delta_{i,j}$ is the Kronecker delta function that is $1$ if $i=j$ and $0$ if $i\neq j$.

Let $x$ be an arbitrary vector in $V$.
Since $\{v_i\}_{i=1}^n$ is a basis of $V$, we express $x\in V$ as a linear combination of the basis. We have
\[x=\sum_{i=1}^nc_iv_i,\] where $c_i$ is a scalar (an element in $k$) for $i=1, \dots, n$.
For a fixed $j$, we have
\begin{align*}
v^j(x)&=v^j \left(\sum_{i=1}^nc_iv_i \right)\\
&=\sum_{i=1}^nc_iv^j(v_i) && \text{ by the linearity of $v_j$}\\
&=\sum _{i=1}^nc_i \delta_{i,j} && \text{ by (*)}\\
&=c_j.
\end{align*}

Thus we have obtained $c_j=v^j(x)$ for any $j$. Substituting this into the linear combination of $x$, we have
\[x=\sum_{i=1}^nv^i(x)v_i\] as required.

Add to solve later

More from my site

Dimension of the Sum of Two Subspaces Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$. Then prove that \[\dim(U+V) \leq \dim(U)+\dim(V).\] Definition (The sum of subspaces). Recall that the sum of subspaces $U$ and $V$ is \[U+V=\{\mathbf{x}+\mathbf{y} \mid […]
Find a Basis and Determine the Dimension of a Subspace of All Polynomials of Degree $n$ or Less Let $P_n(\R)$ be the vector space over $\R$ consisting of all degree $n$ or less real coefficient polynomials. Let \[U=\{ p(x) \in P_n(\R) \mid p(1)=0\}\] be a subspace of $P_n(\R)$. Find a basis for $U$ and determine the dimension of $U$. Solution. […]
Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient. Let $W$ be the following subset of $P_3$. \[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\] Here $p'(x)$ is the first derivative of $p(x)$ and […]
A Basis for the Vector Space of Polynomials of Degree Two or Less and Coordinate Vectors Show that the set \[S=\{1, 1-x, 3+4x+x^2\}\] is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less. Proof. We know that the set $B=\{1, x, x^2\}$ is a basis for the vector space $P_2$. With respect to this basis $B$, the coordinate […]
The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$. Then prove that $V$ is a subspace of $\R^n$. Proof. To prove that $V=\{\mathbf{0}\}$ is a subspace of $\R^n$, we check the following subspace […]
$Linear Transformation and a Basis of the Vector Space $\R^3$$ Linear Transformation and a Basis of the Vector Space $\R^3$ Let $T$ be a linear transformation from the vector space $\R^3$ to $\R^3$. Suppose that $k=3$ is the smallest positive integer such that $T^k=\mathbf{0}$ (the zero linear transformation) and suppose that we have $\mathbf{x}\in \R^3$ such that $T^2\mathbf{x}\neq \mathbf{0}$. Show […]
Isomorphism of the Endomorphism and the Tensor Product of a Vector Space Let $V$ be a finite dimensional vector space over a field $K$ and let $\End (V)$ be the vector space of linear transformations from $V$ to $V$. Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$ be a basis for $V$. Show that the map $\phi:\End (V) \to V^{\oplus n}$ defined by […]
Linear Transformation to 1-Dimensional Vector Space and Its Kernel Let $n$ be a positive integer. Let $T:\R^n \to \R$ be a non-zero linear transformation. Prove the followings. (a) The nullity of $T$ is $n-1$. That is, the dimension of the nullspace of $T$ is $n-1$. (b) Let $B=\{\mathbf{v}_1, \cdots, \mathbf{v}_{n-1}\}$ be a basis of the […]