Short question : Does a basis absolutly has to be square to find the coordinate of a vector in it ?
Detailed question :
First, I am asked to find a base for these 3 vectors.
$\left[ \begin{array}{ccc}
1 & 1 & 4\\
-2 & 1 & 3\\
1 & 1 & 2\\
3 & -2 & 1\end{array} \right]$
Using Gauss, I found that matrix:
$\left[ \begin{array}{ccc}
1 & 1 & 4\\
0 & 3 & 11\\
0 & 0 & -2\\
0 & 0 & 0\end{array} \right]$
my answer : Since I have no free variable, these 3 vectors form a base in $R^4$
Then it says : If possible, find the coordinates of this vector : $[2,−1, 2, 1]$
Then, to find out the coordinates of the vector, I went on and made it an augmented matrix.
$\left[ \begin{array}{ccc|c}
1 & 1 & 4 & 2\\
-2 & 1 & 3 & -1\\
1 & 1 & 2 & 2\\
3 & -2 & 1 & 1\end{array} \right]$
Using Gauss, I found out this matrix
$\left[ \begin{array}{ccc|c}
1 & 1 & 4 & 2\\
0 & 3 & 11 & 3\\
0 & 0 & -2 & 0\\
0 & 0 & 0 & 0\end{array} \right]$
From here I figured that
$
x3 = 0\\ x2 = 1 \\ x1 = 1
$
So I decided to do this linear combination to obtains the coordinate
$ 1 \dot\ v1 + 1 \dot\ v2 + 0 \dot\ v3$
I actually went on and computed the above linear combination and it sucessfully gave me [2,-1,2,1]
So final answer : The coordinates are [c1,c2,c3] = [1,1,0]
but my answer is in $R^3$ ? Is this correct ?
Here is my concern : I think the teacher said that I need a square basis to find the coordinates of a vector within it ?
Best Answer
You're not going to find $3$ vectors in $\mathbb{R}^4$ that span $\mathbb{R}^4$ since that would violate the Dimension Theorem.
Any collection of vectors will span a subspace. In this case there are $3$ linearly independent vectors, so they will be a basis for a $3$-dimensional subspace of $\mathbb{R}^4$.