How can I determine whether a vector can be expressed as a linear combination of a orthonormal set vectors ?
Let's say I have a orthonormal set of vectors $\{v_1, v_2\}$:
$$
v_1=\left(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right)
$$
$$
v_2=\left(\frac{1}{2\sqrt{19}},\:\frac{5}{2\sqrt{19}},\:\frac{1}{2\sqrt{19}},\:\frac{7}{2\sqrt{19}}\right)
$$
$v_1$ and $v_2$ are orthonormal set since $v_1 \cdot v_2 = 0$, and $\|v_1\| = 1, \|v_2\| = 1$.
and I have a vector: $$(1, 1, 1, 1)$$
I know how to express it as linear combination of $v_1$ and $v_2$, but how can I know it can be expressed as a linear combination of $v_1$ and $v_2$ ? Is that enough to just verify the orthonormal set vectors by using the following formula ?
$$
x = (x\cdot v_1)v_1+(x\cdot v_2)v_2
$$
Thank you.
Best Answer
One way is to check the rank of the following matrix: $$A = \begin{bmatrix} -\frac 12 & -\frac 12 & -\frac 12 & \frac 12\\ \frac{1}{2\sqrt{19}} & \frac{5}{2\sqrt{19}} & \frac{1}{2\sqrt{19}} & \frac{7}{2\sqrt{19}} \\ 1 & 1 & 1 & 1 \end{bmatrix} $$
If rank $A = 3$ - which is the case here - then all these $3$ vectors will be linearly indepedent, thus the vector $[1,1,1,1]$ cannot be written as a linear combination of $v_1$ and $v_2$. If rank $A = 2$, then the 3 vectors would have been linearly dependent.