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Express w as a linear combination of the vector V1 and V2 if this is possible. If not, Show that it is impossible.

w= [3]
[-1]
[2]

v1= [-3]
[1]
[2]

v2= [6]
[-2]
[3]

User Elephantik
by
7.1k points

1 Answer

5 votes

Answer:

Explanation:


w=((1)/(7)).V_(1)+((4)/(7)).V_(2)

We have the following vectors :


w=\left[\begin{array}{c}3&-1&2\end{array}\right]


V_(1)=\left[\begin{array}{c}-3&1&2\end{array}\right]


V_(2)=\left[\begin{array}{c}6&-2&3\end{array}\right]

In order to express
w as a linear combination of the vectors
V_(1) and
V_(2), we will search for
a,b ∈ IR such that :


a.V_(1)+b.V_(2)=w (I)

Now we are going to work matrixically with the equation (I) :


a\left[\begin{array}{c}-3&1&2\end{array}\right]+b\left[\begin{array}{c}6&-2&3\end{array}\right]=\left[\begin{array}{c}3&-1&2\end{array}\right]

Distributing mathematically and matching ''component to component'' we lead to the following equations :


-3a+6b=3\\a-2b=-1\\2a+3b=2

Working with the system associated matrix :


\left[\begin{array}{ccc}-3&6&3\\1&-2&-1\\2&3&2\end{array}\right]

Applying matrix operations we lead to the following equivalent matrix :


\left[\begin{array}{ccc}1&0&(1)/(7)\\0&1&(4)/(7)\\0&0&0\end{array}\right]

In this matrix we obtain that :


a=(1)/(7) and
b=(4)/(7)

We can verify this solution by replacing the values of
a and
b in the equation (I) :


((1)/(7)).V_(1)+((4)/(7)).V_(2)=w


((1)/(7)).\left[\begin{array}{c}-3&1&2\end{array}\right]+((4)/(7)).\left[\begin{array}{c}6&-2&3\end{array}\right]=\left[\begin{array}{c}3&-1&2\end{array}\right]


\left[\begin{array}{c}3&-1&2\end{array}\right]=\left[\begin{array}{c}3&-1&2\end{array}\right]

User Troy Weber
by
7.0k points
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