Consider the vector space of functions $y=ax^2 + bx+ c$ for real constants a,b,c. To each function we can associate the vector $(a,b,c)^T$ . Find a basis of the set of these functions that passes through the points (x,y) = (1,2) and (3,4) . Is this set a subspace? What is its dimension?
Here is what I am thinking:
- Solving for the basis:
2 = (1)a + (1)b + c
3 = (4)a + (2)b + c
This can be written in the form of the following matrix:
$
\begin{bmatrix}
1 &4 \\
1 & 2 \\
1 & 1
\end{bmatrix}
$
which can be reduced to:
$\begin{bmatrix}
1 & 0 \\
0 & 1 \\
0 & 0
\end{bmatrix}
$
Thus, our basis is:
$\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}
$ ,
$\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}
$
and it follows that the dimension is 2. I am going in the correct direction?
- How can prove this is a subspace?
Thanks !
No. If you represent a quadratic polynomial by a column vector $\;\smash{\begin{bmatrix}a\\[-0.5ex]b\\[-1ex]c\end{bmatrix}}$,the system of linear equations you obtain is: $$\begin{bmatrix}1&1&1 \\4&2&1\end{bmatrix}\begin{bmatrix}a\\b\\c\end{bmatrix}=\begin{bmatrix}2\\1\end{bmatrix}$$ This is a inhomogeneous linear system. Its solutions are an
affinesubspace, with direction thevectorsubspace of solutions of the associated homogeneous linear system. As the matrix has rank $2$, this subspace has codimension $2$, i.e. dimension $3-2=1$.