I have the following Linear Algebra question:
Prove/disprove: if $(x_1,y_1),(x_2,y_2)$ are 2 linearly independent vectors then $(x_1,y_1,z_1),(x_2,y_2,z_2)$ are also linearly independent vectors
thank you, Dor
I have the following Linear Algebra question:
Prove/disprove: if $(x_1,y_1),(x_2,y_2)$ are 2 linearly independent vectors then $(x_1,y_1,z_1),(x_2,y_2,z_2)$ are also linearly independent vectors
thank you, Dor
Two vectors $(x_1,y_1)$ and $(x_2,y_2)$ are independent if there is no scalar $c$ such that both $cx_1 = x_2$ and $cy_1=y_2$. If these two equations cannot be simultaneously satisfied by a single $c$, then they are independent.
The criterion is basically the same in three dimensions: $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are independent unless there is some scalar $c$ such that $cx_1=x_2$, $cy_1=y_2$, and $cz_1 = z_2$ simultaneously. In your problem, the vectors cannot satisfy the first two equations simultaneously, the certainly cannot satisfy all three. Following this logic, adding components cannot make any number of independent vectors dependent.
Write down the definition of linear independence: $(x_1,y_1,z_1),(x_2,y_2,z_2) {\mbox{ are linearly independent}}\Leftrightarrow\left(\forall\alpha,\beta\hspace{3pt} \alpha(x_1,y_1,z_1)+\beta(x_2,y_2,z_2)=0\rightarrow\alpha=\beta=0\right)$ Now expand it to the coordinates. If $\alpha(x_1,y_1,z_1)+\beta(x_2,y_2,z_2)=0$ then, in particular $\alpha(x_1,y_1)+\beta(x_2,y_2)=0$. What does this imply?