(a) Prove that the set $\{ 1 , 1 + x , (1 + x)^2 \}$ is a basis for $\mathbf{P}_2$.
Consider the standard basis $\mathfrak{B} = \{ 1 , x , x^2 \}$ of $\mathbf{P}_2$. Using this basis, we can write the elements using coordinate vectors as
\[ [1]_{\mathfrak{B}} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \quad [1+x]_{\mathfrak{B}} = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \quad [(1+x)^2]_{\mathfrak{B}} = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}.\]
We find the coordinate vector by writing an element as a linear combination of the basis elements. For example, $(1+x)^2 = 1 + 2x + 1 x^2$, and so the coefficients $1, 2, 1$ translate into the column vector $ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$.
Because $\dim \mathbf{P}_2 = 3$, this set is a basis if and only if these three vectors are linearly independent. To verify this, consider the matrix
\[\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}.\]
This matrix is upper-triangular, and the diagonal entries are all non-zero. This implies the matrix is non-singular, and so the columns are linearly independent.
Thus, the set $\{ 1 , 1+x , (1+x)^2 \}$ is a basis of $\mathbf{P}_2$.
(b) Write the polynomial $f(x) = 2 + 3x – x^2$ as a linear combination of the basis $\{ 1 , 1+x , (1+x)^2 \}$.
First Method: Using the technique of completing the square, we can factor the polynomial $f(x)$ as we like. Specifically,
\begin{align*}
f(x) &= -x^2 + 3x + 2 \\
&= – (x+1)^2 + 5x + 3 \\
&= – (x+1)^2 + 5(x+1) – 2.
\end{align*}
Hence, we have the linear combination
\[f(x)= -2\cdot 1 +5(1+x) -(1+x)^2. \tag{*}\]
Second Method: We can find this factorization by calculating the Taylor polynomial of $f(x)$ centered at $-1$.
This Taylor polynomial is defined by
\[f(x) = f(-1) + f'(-1) (x+1) + \frac{ f”(-1)}{2} (x+1)^2.\]
The polynomial in (*) is recovered by finding $f(-1) = -2$, $f'(-1) = 5$, and $f^{\prime \prime}(-1) = -2$.
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\[B = \left\{ 1+x , 1+x^2 , x - x^2 + 2x^3 , 1 - x - x^2 \right\}.\]
Write the coordinate vector for the polynomial $f(x) = -3 + 2x^3$ in terms of the basis […]
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Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$.
Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of […]
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Determine whether each of the following sets is a basis for $\R^3$.
(a) $S=\left\{\, \begin{bmatrix}
1 \\
0 \\
-1
\end{bmatrix}, \begin{bmatrix}
2 \\
1 \\
-1
\end{bmatrix}, \begin{bmatrix}
-2 \\
1 \\
4
\end{bmatrix} […]
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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 […]
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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 $P_3$.
\[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\]
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For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.
Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, […]