Vector quantity Proof

Question

(a) Show that any two-dimensional vector can be expressed in the form $$s\begin{pmatrix} 3 \\ -1 \end{pmatrix} + t\begin{pmatrix} 2 \\ 7 \end{pmatrix},$$where $s$ and $t$ are real numbers.

(b) Let $u$ and $v$ be non-zero vectors. Show that any two-dimensional vector can be expressed in the form $$su + tv,$$ where $s$ and $t$ are real numbers, if and only if of the vectors $u$ and $v$, one vector is not a scalar multiple of the other vector.

For these two equations I have thought about using a system of equations and graph but it does not bring me anywhere. So far I have $$3s+2t = -s+7t \implies s = \frac{5}{4}t$$

Answer 1

You need to solve

$$ s\begin{pmatrix} 3 \\ -1 \end{pmatrix} + t\begin{pmatrix} 2 \\ 7 \end{pmatrix}= \begin{pmatrix} a \\ b \end{pmatrix}$$ for $s$ and $t$ in terms of $a$ and $b$

For the second part of the question you need to solve $$ s\begin{pmatrix} u_1\\ u_2 \end{pmatrix} + t\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}= \begin{pmatrix} a \\ b \end{pmatrix}$$ for $s$ and $t$ in terms of $u_1,u_2$ and $v_1,v_2$ and $a,b$

The condition on vectors $u$ and $v$ make it possible to solve this system for s and t.

Answer 2

Take a vector $(a,b)$. You want to prove that there exists $s,t$ such that $$(a,b) = s(3,-1) + t(2,7)$$

This is equivalent to the system $$a = 3s + 2t \quad \text{ and } \quad b = -s + 7t$$

Now you just have to solve this system to see that it has a unique solution for every $a,b$.

Answer 3

For (a), you need to show that you can write any vector $(x,y)$ in the form described.That is, you need to show that there exists a solution $(s,t)$ to the equation $$s\begin{pmatrix} 3 \\ -1 \end{pmatrix} + t\begin{pmatrix} 2 \\ 7 \end{pmatrix}=\begin{pmatrix} x \\ y \end{pmatrix}$$ This equation can also be written as $$\begin{pmatrix} 3 & 2 \\ -1 & 7 \end{pmatrix}\begin{pmatrix} s \\ t \end{pmatrix}=\begin{pmatrix} x \\ y \end{pmatrix}$$ You need to show that there is a solution (but do not actually need to find it). (Look at the determinant of the matrix, or the independence of its columns.)

Part (b) can be answered in a very similar way.

Answer 4

You just need to prove that the two vectors are independent.

$$s\vec{u}+t\vec{v}=\vec{0}$$

$$\implies 3s+2t=0\; ; \; -s+7t=0$$

$$\implies 2t+21t=0\implies t=s=0$$ Done.

You can also use the deteminant

$$det(\vec{u},\vec{v})=3.7-(-1).2=23\ne 0$$

thus they are independant.