# given 2 linearly independent vectors: $(x_1,y_1),(x_2,y_2)$ prove that $(x_1,y_1,z_1),(x_2,y_2,z_2)$ are also independent

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

• Write out a generic linear combination of these 3d vectors. What do you get in the first two coordinates? How does it relate to your 2d vectors?
– WimC
Nov 30, 2012 at 11:42
• I don't think i completely understand. I know that a*(x1,y1)+b*(x2,y2) = 0 only if both a,b = 0. But does that necessarily means that a*(x1,y1,z1)+b*(x2,y2,z2) =0 also only if both a,b = 0? Nov 30, 2012 at 11:50

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.
• In layman's terms: a pair of vectors is independent if they are not parallel. Adding $z$ components to two non-parallel plane vectors cannot make them parallel in three dimensions. Dec 1, 2012 at 1:33
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?
• first of all, thank you all for your help, I am new here... so first let my correct my question, i'm not sure if it matters: prove/disprove: \begin{pmatrix} x1 \\ y1 \end{pmatrix} and \begin{pmatrix} x2 \\ y2 \end{pmatrix} are linearly independent --> \begin{pmatrix} x1 \\ y1 \\ z1 \end{pmatrix},\begin{pmatrix} x2 \\ y2 \\ z2 \end{pmatrix} are independent. i'm not sure if i can say that z1+z2 = 0 only if $\alpha = \beta = 0$ Nov 30, 2012 at 12:15