Linear Algebra

Find a Basis and Determine the Dimension of a Subspace of All Polynomials of Degree $n$ or Less

Problem 137

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 $U$.

Add to solve later

Solution.

Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be an element in $U$.
Since $p(1)=0$, we have $a_n+a_{n-1}+\cdots+a_1+a_0=0$. Thus we have
\begin{align*}
p(x)&=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x-(a_n+a_{n-1}+\cdots+a_1)\\
&=a_n(x^n-1)+a_{n-1}(x^{n-1}-1)+\cdots a_1(x-1).
\end{align*}

Let us define
\[q_n(x)=x^n-1, q_{n-1}(x)=x^{n-1}-1, \dots, q_1(x)=x-1 \in U.\] Then the above computation shows that we have
\[p(x)=a_nq_n(x)+a_{n-1}q_{n-1}(x)+\cdots a_1q_1(x).\] In other words, any element $p(x)\in U$ is a linear combination of $q_1(x), \dots, q_n(x)$.
Hence $B:=\{q_1(x), \dots, q_n(x)\}$ is a spanning set for $U$.

We show that $B$ is a linearly independent set.
Consider a linear combination
\[c_1q_1(x)+\cdots+c_n q_n(x)=\theta(x):=0x^{n}+0x^{n-1}+\cdots+0x+0.\]

Then we have
\begin{align*}
c_nx^n+c_{n-1}x^{n-1}+\cdots+c_1x-(c_1+c_2+\cdots+c_n)=0.
\end{align*}
Comparing the coefficients of both sides, we obtain
\[c_1=c_2=\cdots=c_n=0\] and thus $q_1(x), \dots, q_n(x)$ are linearly independent, and hence $B=\{q_1(x), \dots, q_n(x)\}$ is a basis for the subspace $U$. Since the dimension is the number of vectors in a basis, the dimension of $U$ is $n$.

Add to solve later

More from my site

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 […]
Linear Dependent/Independent Vectors of Polynomials Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent? (a) At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ […]
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 […]
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 […]
Prove a Given Subset is a Subspace and Find a Basis and Dimension Let \[A=\begin{bmatrix} 4 & 1\\ 3& 2 \end{bmatrix}\] and consider the following subset $V$ of the 2-dimensional vector space $\R^2$. \[V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.\] (a) Prove that the subset $V$ is a subspace of $\R^2$. (b) Find a basis for […]
Vector Space of Polynomials and Coordinate Vectors Let $P_2$ be the vector space of all polynomials of degree two or less. Consider the subset in $P_2$ \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} &p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\ &p_3(x)=2x^2, &p_4(x)=2x^2+x+1. \end{align*} (a) Use the basis […]
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 […]