Engineering Mathematics
Linear Combinations &Systems of Linear Equations
Example1
Finding the value of k for a linear combination
Problem
For which values of \(k\) will the vector \(v = (1,\,-2,\,k)\) in \(\mathbb{R}^3\) be a linear combination of the vectors \(u = (3,\,0,\,-2)\) and \(w = (2,\,-1,\,-5)\)?
Given
\(u = (3,\,0,\,-2)\)
\(w = (2,\,-1,\,-5)\)
\(v = (1,\,-2,\,k)\)
Suppose \(v\) is a linear combination of \(u\) and \(w\). Then there exist scalars \(x, y\) such that \[xu + yw = v\] That is, \[x(3,\,0,\,-2) + y(2,\,-1,\,-5) = (1,\,-2,\,k).\]
System of equations
| \(3x + 2y\) | \(= 1\) | — (1) |
| \(-y\) | \(= -2\) | — (2) |
| \(-2x - 5y\) | \(= k\) | — (3) |
Solution steps
Step 1 — Solve equations (1) and (2)
From equation (2): \(-y = -2 \;\Rightarrow\; y = 2\).
Substitute into (1): \(3x + 2(2) = 1 \;\Rightarrow\; 3x = -3 \;\Rightarrow\; x = -1\).
Result: x = −1 and y = 2
Substitute into (1): \(3x + 2(2) = 1 \;\Rightarrow\; 3x = -3 \;\Rightarrow\; x = -1\).
Result: x = −1 and y = 2
Step 2 — Substitute into equation (3) to find k
\[
-2x - 5y = k
\]
\[
-2(-1) - 5(2) = k
\]
\[
2 - 10 = k
\]
\[
k = -8
\]
The vector \(v\) is a linear combination of \(u\) and \(w\) only if \(\boldsymbol{k = -8}\).
That is, \(v = (-1)\,u + 2\,w\) when \(k = -8\).
That is, \(v = (-1)\,u + 2\,w\) when \(k = -8\).
Tutorial Video: Linear Combination & System of Equations
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