Math Problem Statement

If u = (2, 4, 2), v = (-2, 1, 5), and w = (5, 1, 0), calculate the following linear combination: 2u + 3v + 2w.

Solution

I can see the image you uploaded, which contains a linear combination problem. Let's solve it step by step.

Given:

\mathbf{v} = \begin{pmatrix} -2 \\ 1 \\ 5 \end{pmatrix}, \mathbf{w} = \begin{pmatrix} 5 \\ 1 \\ 0 \end{pmatrix}$$ You are asked to compute: $$2\mathbf{u} + 3\mathbf{v} + 2\mathbf{w}$$ First, let's compute each term: 1. $$2\mathbf{u} = 2 \times \begin{pmatrix} 2 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 4 \\ 8 \\ 4 \end{pmatrix}$$ 2. $$3\mathbf{v} = 3 \times \begin{pmatrix} -2 \\ 1 \\ 5 \end{pmatrix} = \begin{pmatrix} -6 \\ 3 \\ 15 \end{pmatrix}$$ 3. $$2\mathbf{w} = 2 \times \begin{pmatrix} 5 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 10 \\ 2 \\ 0 \end{pmatrix}$$ Now, summing these vectors: $$\begin{pmatrix} 4 \\ 8 \\ 4 \end{pmatrix} + \begin{pmatrix} -6 \\ 3 \\ 15 \end{pmatrix} + \begin{pmatrix} 10 \\ 2 \\ 0 \end{pmatrix} = \begin{pmatrix} (4 - 6 + 10) \\ (8 + 3 + 2) \\ (4 + 15 + 0) \end{pmatrix} = \begin{pmatrix} 8 \\ 13 \\ 19 \end{pmatrix}$$ Thus, the result of the linear combination is: $$\mathbf{r} = \begin{pmatrix} 8 \\ 13 \\ 19 \end{pmatrix}$$ Would you like further details on this process or have any questions? Here are 5 related questions to expand this topic: 1. What is a general definition of a linear combination in vector spaces? 2. How can linear independence of vectors be determined? 3. Can you compute a different linear combination using scalar multiples? 4. What role do linear combinations play in solving systems of linear equations? 5. How does this relate to the span of vectors? **Tip**: When solving vector operations, always ensure each term is correctly multiplied before performing any addition or subtraction.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Operations

Formulas

Scalar Multiplication: c * (a, b, c) = (ca, cb, cc)
Vector Addition: (a1, b1, c1) + (a2, b2, c2) = (a1 + a2, b1 + b2, c1 + c2)

Theorems

-

Suitable Grade Level

Grades 9-12

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