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Chapter 1: Problem 10

Compute the given linear combination of \(\mathbf{u}=[1,2,1,0], \mathbf{v}=[-2,0,1,6]\), and \(\mathbf{w}=[3,-5,1,-2]\). $$3 \mathrm{u}+\mathrm{v}-\mathrm{w}$$

Short Answer

Expert verified
The result of the linear combination is \([-2, 11, 3, 8]\).

Step by step solution

01

Distribute Scalar to Vector \( \mathbf{u} \)

Multiply each component of vector \( \mathbf{u} = [1, 2, 1, 0]\) by the scalar 3. \(3 \mathbf{u} = 3 \times [1, 2, 1, 0] = [3, 6, 3, 0]\)
02

Multiply Vector \( \mathbf{v} \) by 1

Since the expression \( +\mathbf{v} \) is equivalent to multiplying \( \mathbf{v} \) by 1, simply keep vector \( \mathbf{v} \) unchanged. \( \mathbf{v} = [-2, 0, 1, 6]\)
03

Calculate Negative of Vector \( \mathbf{w} \)

Multiply each component of vector \( \mathbf{w} = [3, -5, 1, -2]\) by -1 to find \( -\mathbf{w} \). \(-\mathbf{w} = -1 \times [3, -5, 1, -2] = [-3, 5, -1, 2]\)
04

Add the Vectors Together

Add the results from Steps 1, 2, and 3 component-wise: \[3 \mathbf{u} + \mathbf{v} - \mathbf{w} = [3, 6, 3, 0] + [-2, 0, 1, 6] + [-3, 5, -1, 2].\]Add each corresponding component: - \((3) + (-2) + (-3) = -2\)- \((6) + (0) + (5) = 11\)- \((3) + (1) + (-1) = 3\)- \((0) + (6) + (2) = 8\)
05

Final Step: Write the Result

Combine the sums from Step 4 into a single vector: \([-2, 11, 3, 8]\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Addition
Vector addition is a fundamental concept in linear algebra, where we combine two or more vectors to produce a new vector. This process involves adding each pair of corresponding components from the vectors. Imagine you have two vectors, \( \mathbf{a} = [a_1, a_2, a_3] \) and \( \mathbf{b} = [b_1, b_2, b_3] \). To add these vectors, you perform the addition component-wise:
  • First component: \( a_1 + b_1 \)
  • Second component: \( a_2 + b_2 \)
  • Third component: \( a_3 + b_3 \)
This results in a new vector \( \mathbf{c} = [a_1+b_1, a_2+b_2, a_3+b_3] \). Vector addition is both associative and commutative, which means you can add vectors in any order, and the sum will be the same. Understanding this will help in efficiently combining multiple vectors to solve complex problems.
Scalar Multiplication
Scalar multiplication involves stretching or compressing a vector by a scalar (a single number). Think of a vector \( \mathbf{v} = [v_1, v_2, v_3] \). When you multiply it by a scalar \( k \), you multiply each component of the vector by \( k \), resulting in a new vector:
  • First component: \( k \times v_1 \)
  • Second component: \( k \times v_2 \)
  • Third component: \( k \times v_3 \)
This gives you the new vector \( k \mathbf{v} = [kv_1, kv_2, kv_3] \). Scalar multiplication changes the magnitude of the vector but does not affect its direction unless you multiply by a negative scalar, which would also reverse the direction. This operation is vital in transformations and scaling within various application contexts.
Linear Algebra Problem
Linear algebra problems often involve operations like vector addition and scalar multiplication. These operations are used to solve a range of problems including systems of equations, projections, and transformations. In a typical linear algebra problem, such as computing a linear combination of vectors, you might need to work with expressions like \( 3\mathbf{u} + \mathbf{v} - \mathbf{w} \). To solve such problems, break down each term into simpler operations:
  • Apply scalar multiplication: Scale vectors by appropriate scalars.
  • Use vector addition/subtraction: Add or subtract vectors component-wise.
Through these steps, you can calculate the final resulting vector. These basic operations build the foundation for more advanced topics, like finding vector spaces, basis, and dimensions.
Coordinate Calculation
Coordinate calculation involves determining the specific components of a vector as a result of linear algebra operations. In our example problem, we are computing \( 3\mathbf{u} + \mathbf{v} - \mathbf{w} \). Each vector has components represented as coordinates which need to be precisely calculated to determine the final vector:
  • Step 1: Calculate \( 3\mathbf{u} \), multiplying each component by 3.
  • Step 2: Consider \( \mathbf{v} \), which remains unchanged.
  • Step 3: Find \(-\mathbf{w} \) by reversing the sign of each component.
  • Step 4: Add results for each component from the previous steps to find the final coordinates.
This step-by-step process shows how careful calculation and combination of coordinates help find the specific vector that solves our problem. Coordinate calculation is vital for precision in fields like physics or engineering where exact measurements are crucial.

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Most popular questions from this chapter

In Exercises \(1-17\), let \(\mathrm{u}=[-1,3,4], \mathrm{v}=\) \([2,1,-1]\), aid \(w=[-2,-1,3]\). Find the indicated quantity. $$ \|v\| $$

In Exercises \(1-17\), let \(\mathrm{u}=[-1,3,4], \mathrm{v}=\) \([2,1,-1]\), aid \(w=[-2,-1,3]\). Find the indicated quantity. The value of \(x\) such that \([x,-3,5]\) is perpendicular to u

Write the linear system $$ \begin{array}{r} 3 x-2 y+4 z=10 \\ x-y-3 z=0 \\ 2 x+y-5 z=-3 \end{array} $$ as a column-vector equation.

Determine whether the given transition matrix with the indicated distribution of zero entries and nontero \(x\) entries is regular. $$\left[\begin{array}{cccc}0 & 0 & x & x \\ x & 0 & 0 & x \\ 0 & x & 0 & x \\\ 0 & 0 & 0 & x\end{array}\right]$$

In Exercises \(1-17\), let \(\mathrm{u}=[-1,3,4], \mathrm{v}=\) \([2,1,-1]\), aid \(w=[-2,-1,3]\). Find the indicated quantity. \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})\)
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