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Chapter 1: Problem 26

For a fixed \(a \in R\), determine the dimension of the subspace of \(\mathrm{P}_{n}(R)\) defined by $\left\\{f \in \mathrm{P}_{n}(R): f(a)=0\right\\}$.

Short Answer

Expert verified
The dimension of the subspace of \(\mathrm{P}_{n}(R)\) defined by \(\{f \in \mathrm{P}_{n}(R): f(a)=0\}\) is \(n\), and its basis is given by \(B = \{ (x-a), (x-a)^2, \dots, (x-a)^n\}\).

Step by step solution

01

Understand the given subspace

The exercise states that we want to find the dimension of a subspace of \(\mathrm{P}_{n}(R)\), which is the vector space of all real polynomials with degree less than or equal to \(n\). The subspace is defined by the set of all polynomials \(f\) such that \(f(a) = 0\), for some fixed real number \(a\). This means that the polynomials in this subspace must satisfy the condition that when you plug in \(a\) into the polynomial, the result is 0.
02

Identify a basis for the subspace

To find the dimension of the subspace, we need to find a basis for it. A basis for the subspace would be a set of linearly independent polynomials that span the subspace. We can think of a polynomial in \(\mathrm{P}_{n}(R)\) as \[ f(x) = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots + c_n(x-a)^n, \] where \(c_i \in R\), and \(i = 0, 1, \dots, n\). Notice that when we plug in \(x = a\), we get \[ f(a) = c_0 + c_1(a-a) + c_2(a-a)^2 + \cdots + c_n(a-a)^n = c_0, \] hence \(c_0 = 0\). This means that the polynomials in the subspace are of the form \[ g(x) = c_1(x-a) + c_2(x-a)^2 + \cdots + c_n(x-a)^n. \] We can take the set \(B = \{ (x-a), (x-a)^2, \dots, (x-a)^n\}\) as a basis of the subspace, as these polynomials are linearly independent and span the subspace.
03

Compute the dimension of the subspace

The dimension of a subspace is simply the cardinality (number of elements) in its basis. In this case, the basis \(B\) contains \(n\) elements, therefore the dimension of the subspace is \(n\).

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Most popular questions from this chapter

Label the following statements as true or false. (a) If \(V\) is a vector space and \(W\) is a subset of \(V\) that is a vector space, then \(W\) is a subspace of \(V\). (b) The empty set is a subspace of every vector space. (c) If \(V\) is a vector space other than the zero vector space, then \(V\) contains a subspace \(W\) such that \(W \neq V\). (d) The intersection of any two subsets of \(V\) is a subspace of \(V\). (e) An \(n \times n\) diagonal matrix can never have more than \(n\) nonzero entries. (f) The trace of a square matrix is the product of its diagonal entries. (g) Let \(\mathrm{W}\) be the \(x y\)-plane in \(\mathrm{R}^{3}\); that is, $\mathrm{W}=\left\\{\left(a_{1}, a_{2}, 0\right): a_{1}, a_{2} \in R\right\\}\(. Then \)W=R^{2}$.

Let $\mathrm{V}=\left\\{\left(a_{1}, a_{2}\right): a_{1}, a_{2} \in F\right\\}\(, where \)F\( is a field. Define addition of elements of \)\mathrm{V}$ coordinatewise, and for \(c \in F\) and $\left(a_{1}, a_{2}\right) \in \mathrm{V}$, define $$ c\left(a_{1}, a_{2}\right)=\left(a_{1}, 0\right) . $$ Is \(\mathrm{V}\) a vector space over \(F\) with these operations? Justify your answer.

In each part, determine whether the given vector is in the span of \(S\). (a) \((2,-1,1), \quad S=\\{(1,0,2),(-1,1,1)\\}\) (b) \((-1,2,1), \quad S=\\{(1,0,2),(-1,1,1)\\}\) (c) \((-1,1,1,2), \quad S=\\{(1,0,1,-1),(0,1,1,1)\\}\) (d) \((2,-1,1,-3), \quad S=\\{(1,0,1,-1),(0,1,1,1)\\}\) (e) $-x^{3}+2 x^{2}+3 x+3, \quad S=\left\\{x^{3}+x^{2}+x+1, x^{2}+x+1, x+1\right\\}$ (f) $2 x^{3}-x^{2}+x+3, \quad S=\left\\{x^{3}+x^{2}+x+1, x^{2}+x+1, x+1\right\\}$ (g) $\left(\begin{array}{rr}1 & 2 \\ -3 & 4\end{array}\right), \quad S=\left\\{\left(\begin{array}{rr}1 & 0 \\ -1 & 0\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 0 & 1\end{array}\right),\left(\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right)\right\\}$ (h) $\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right), \quad S=\left\\{\left(\begin{array}{rr}1 & 0 \\ -1 & 0\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 0 & 1\end{array}\right),\left(\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right)\right\\}$

Show that the matrices $$ \left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right), \quad\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right), \quad\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right) $$ generate \(\mathrm{M}_{2 \times 2}(F)\).

Show that \(P_{n}(F)\) is generated by \(\left\\{1, x, \ldots, x^{n}\right\\}\).
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