Hi can someone help me with this question:
Find a vector in $\mathbb{R}^5$ which is not a linear combination of u and v. Verify that your vector is not a linear combination of u and v. Where u = (1,0,-1,1,1) and v = (3,-2,-1,1,-1)
I got (1,2,3,4,5) is this correct??
Thanks
On
Make two vectors in $\mathbb R^3$ by taking the first three components of each vector:
$$\hat A = (1, 0, -1) \text{ and } \hat B = (3, -2, -1)$$
$\hat A$ and $\hat B$ are clearly linearly independent
Hence the cross product
$$\hat C = \hat A \times\hat B = (-2,-2,-2)$$
is orthogonal to $\hat A$ and $\hat B$. Hence $\hat C$ it is not a linear combination of $\hat A$ and $\hat B$.
It follows that $C = (-2,-2,-2,x,y)$ is not a linear combination of $A$ and $B$ for any $x$ and $y$.
In general $\hat C = (a,b,c)$ can be any $a,b,$ and $c$ such that $$ \left \lVert \begin{matrix} 1 & 0 & -1\\ 3 & -2 & -1\\ a & b & c\\ \end{matrix} \right \rVert \ne 0$$
The vector you need is: $(5,-2,-2,2,0)$. For if it is a linear combination of the other two, then for some real numbers $x$ and $y$ you have:
$(5,-2,-2,2,0) = x(3,-2,-1,1,-1) + y(1,0,-1,1,1)$. So $0 = -x + y$, and $2 = x + y$. So $x = y = 1$, but then $3x + y = 3\cdot 1 + 1 = 4 \neq 5$. So this vector cannot be a linear combination of the given vectors.