Problem 39 Write $\mathbf{v}$ as a linear... [FREE SOLUTION]

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Elementary Linear Algebra

Ron Larson, David C. Falvo

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 4: Problem 39

Write $\mathbf{v}$ as a linear combination of $\mathbf{u}$ and $\mathbf{w},$ if possible, where $\mathbf{u}=(1,2)$ and $\mathbf{w}=(1,-1)$. $$\mathbf{v}=(2,1)$$

Short Answer

Expert verified

The vector $\mathbf{v}=(2,1)$ can be written as a linear combination of $\mathbf{u}=(1,2)$ and $\mathbf{w}=(1,-1)$ as follows: $\mathbf{v}= 1\mathbf{u} + 1\mathbf{w}$.

Step by step solution

Set up the equations

We need to find constants $a$ and $b$ such that $a\mathbf{u} + b\mathbf{w} = \mathbf{v}$. This means we are trying to find $a$ and $b$ to satisfy $(a, 2a) + (b, -b) = (2,1)$. This results into a system of two equations: $a + b = 2$ and $2a - b = 1$.

Solve the system of equations

We can start by adding the two equations, which results into $3a = 3$. Solving for $a$, we find $a = 1$. Substituting $a = 1$ into the first equation, we then get $1 + b = 2$, which gives $b = 1$.

Write the solution as a linear combination

So, with $a = 1$ and $b = 1$, $\mathbf{v}$ can be written as equal to $1\mathbf{u} + 1\mathbf{w}$ or in other words, $\mathbf{v}$ is a linear combination of $\mathbf{u}$ and $\mathbf{w}$.

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91影视

Write \(\mathbf{v}\) as a linear combination of \(\mathbf{u}\) and \(\mathbf{w},\) if possible, where \(\mathbf{u}=(1,2)\) and \(\mathbf{w}=(1,-1)\). $$\mathbf{v}=(2,1)$$

Short Answer

Step by step solution

Set up the equations

Solve the system of equations

Write the solution as a linear combination

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