Linear Combination of Vectors
N/D-25-R-21 | Worked Example
Tutorial Video: Linear Combination & System of Equations
Is it possible to write the vector \((-2,\,0,\,3)\) as a linear combination of the two vectors \((1,3,0)\) and \((2,4,-1)\)? Justify.
Solution
Given
\( u = (1,3,0), \quad v = (2,4,-1), \quad w = (-2,0,3) \)
Suppose \(w\) is a linear combination of \(u\) and \(v\). Then there exist scalars \(x, y\) such that \[ xu + yv = w. \] That is, \[ x(1,3,0) + y(2,4,-1) = (-2,\,0,\,3). \] Comparing components we get the system:
| \( x + 2y = -2 \) | — (1) |
| \( 3x + 4y = \phantom{-}0 \) | — (2) |
| \( \phantom{3x +{}} {-y} = \phantom{-}3 \) | — (3) |
Solving equations (1) and (2) simultaneously:
From (1): \( x = -2 - 2y \)
Substitute into (2): \( 3(-2-2y)+4y=0 \;\Rightarrow\; -6-6y+4y=0 \;\Rightarrow\; y=-3 \)
Back-substitute: \( x = -2 - 2(-3) = -2+6 = 4 \)
\( \therefore\; x = 4, \quad y = -3 \)
Verify in equation (3)
\( -y = -(-3) = 3 \;\checkmark \)
Equation (3) is satisfied.
Conclusion
Yes, \(w\) is a linear combination of \(u\) and \(v\).
Specifically, \( 4u - 3v = w \).
Specifically, \( 4u - 3v = w \).
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