# Basis of Span in Vector Space of Polynomials of Degree 2 or Less

## Problem 367

Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients.

Let

\[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\]
be the set of four vectors in $P_2$.

Then find a basis of the subspace $\Span(S)$ among the vectors in $S$.

(*Linear Algebra Exam Problem, the Ohio State University*)

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## Solution.

Let $B=\{1, x, x^2\}$ be the standard basis of the vector space $P_2$.

With respect to the basis $B$, the coordinate vectors of vectors in $S$ are

\[[1+x+2x^2]_B=\begin{bmatrix}

1 \\

1 \\

2

\end{bmatrix}, \quad [x+2x^2]_B=\begin{bmatrix}

0 \\

1 \\

2

\end{bmatrix}, \quad [-1]_B=\begin{bmatrix}

-1 \\

0 \\

0

\end{bmatrix}, \quad [x^2]_B=\begin{bmatrix}

0 \\

0 \\

1

\end{bmatrix}.\]
Let

\[T=\left\{\, [1+x+2x^2]_B, [x+2x^2]_B, [-1]_B, [x^2]_B \,\right\}\]
be the set of these coordinate vectors.

We then find a basis of $\Span(T)$ among vectors in $T$ by the leading 1 method.

We reduce the augmented matrix by elementary row operations as follows. We have

\begin{align*}

\begin{bmatrix}

1 & 0 & -1 & 0 \\

1 &1 & 0 & 0 \\

2 & 2 & 0 & 1

\end{bmatrix}

\xrightarrow{R_3-2R_2}

\begin{bmatrix}

1 & 0 & -1 & 0 \\

1 &1 & 0 & 0 \\

0 & 0 & 0 & 1

\end{bmatrix}

\xrightarrow{R_2-R_1}

\begin{bmatrix}

1 & 0 & -1 & 0 \\

0 &1 & 1 & 0 \\

0 & 0 & 0 & 1

\end{bmatrix}.

\end{align*}

The last matrix is in reduced row echelon form and the first, the second, and the fourth columns contain the leading 1’s.

Therefore, it follows that

\[\left\{\, [1+x+2x^2]_B, [x+2x^2]_B, [x^2]_B \,\right\}\]
is a basis of $\Span(T)$, and hence

\[\{1+x+2x^2, \quad x+2x^2, \quad x^2\}\]
is a basis of $\Span(S)$ consisting of the vectors of $S$.

## Linear Algebra Midterm Exam 2 Problems and Solutions

- True of False Problems and Solutions: True or False problems of vector spaces and linear transformations
- Problem 1 and its solution: See (7) in the post “10 examples of subsets that are not subspaces of vector spaces”
- Problem 2 and its solution: Determine whether trigonometry functions $\sin^2(x), \cos^2(x), 1$ are linearly independent or dependent
- Problem 3 and its solution: Orthonormal basis of null space and row space
- Problem 4 and its solution (current problem): Basis of span in vector space of polynomials of degree 2 or less
- Problem 5 and its solution: Determine value of linear transformation from $R^3$ to $R^2$
- Problem 6 and its solution: Rank and nullity of linear transformation from $R^3$ to $R^2$
- Problem 7 and its solution: Find matrix representation of linear transformation from $R^2$ to $R^2$
- Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space

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