Question

In each case, either express \(\mathbf{y}\) as a linear combination of \(\mathbf{a}_{1}, \mathbf{a}_{2},\) and \(\mathbf{a}_{3},\) or show that it is not such a linear combination. Here: $$ \begin{array}{l} \mathbf{a}_{1}=\left[\begin{array}{r} -1 \\ 3 \\ 0 \\ 1 \end{array}\right], \mathbf{a}_{2}=\left[\begin{array}{l} 3 \\ 1 \\ 2 \\ 0 \end{array}\right], & \text { and } \mathbf{a}_{3}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array}\right] \\ \text { a. } \mathbf{y}=\left[\begin{array}{l} 1 \\ 2 \\ 4 \\ 0 \end{array}\right] & \text { b. } \mathbf{y}=\left[\begin{array}{r} -1 \\ 9 \\ 2 \\ 6 \end{array}\right] \end{array} $$

Short answer

Case a: Yes, \( \mathbf{y} \) is expressible. Case b: Yes, it is expressible.

Step-by-step solution

Set Up the System for Case a

To determine if \( \mathbf{y} = \begin{bmatrix} 1 \ 2 \ 4 \ 0 \end{bmatrix} \) can be expressed as a linear combination of \( \mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3} \), write the equation: \( \mathbf{y} = c_{1} \mathbf{a}_{1} + c_{2} \mathbf{a}_{2} + c_{3} \mathbf{a}_{3} \). This results in the system of equations: \[ \begin{align*} -c_{1} + 3c_{2} + c_{3} &= 1 \ 3c_{1} + c_{2} + c_{3} &= 2 \ 2c_{2} + c_{3} &= 4 \ c_{1} + c_{3} &= 0 \end{align*} \]

Solve the System for Case a

Substitute \( c_{1} = -c_{3} \) from the last equation into the others. Substitute into the first:\( -(-c_{3}) + 3c_{2} + c_{3} = 1 \Rightarrow c_{3} + 3c_{2} = 1 \). Substitute into the second:\( 3(-c_{3}) + c_{2} + c_{3} = 2 \Rightarrow -3c_{3} + c_{2} + c_{3} = 2 \Rightarrow c_{2} = 2 + 2c_{3} \). Substitute into the third:\( 2c_{2} + c_{3} = 4 \Rightarrow 2(2 + 2c_{3}) + c_{3} = 4 \Rightarrow 4 + 4c_{3} + c_{3} = 4 \Rightarrow 5c_{3} = 0 \Rightarrow c_{3} = 0 \). With \( c_{3} = 0 \), then \( c_{1} = 0 \) and \( c_{2} = 2 \). Verify by substituting those values back to original equations.

Conclusion for Case a

Given that the system has a solution \( c_1 = 0, c_2 = 2, c_3 = 0 \), \( \mathbf{y} = c_{1} \mathbf{a}_{1} + c_{2} \mathbf{a}_{2} + c_{3} \mathbf{a}_{3} \) can be expressed as a linear combination.

Set Up the System for Case b

To determine if \( \mathbf{y} = \begin{bmatrix} -1 \ 9 \ 2 \ 6 \end{bmatrix} \) can be expressed as a linear combination, set up the equation \( \mathbf{y} = c_{1} \mathbf{a}_{1} + c_{2} \mathbf{a}_{2} + c_{3} \mathbf{a}_{3} \), resulting in:\[ \begin{align*} -c_{1} + 3c_{2} + c_{3} &= -1 \ 3c_{1} + c_{2} + c_{3} &= 9 \ 2c_{2} + c_{3} &= 2 \ c_{1} + c_{3} &= 6 \end{align*} \]

Solve the System for Case b

From the last equation, \( c_{1} = 6 - c_{3} \). Substitute into the first:\( -(6-c_{3}) + 3c_{2} + c_{3} = -1 \Rightarrow -6 + 3c_{2} + 2c_{3} = -1 \Rightarrow 3c_{2} + 2c_{3} = 5 \). Substitute into the second:\( 3(6 - c_{3}) + c_{2} + c_{3} = 9 \Rightarrow 18 - 3c_{3} + c_{2} + c_{3} = 9 \Rightarrow c_{2} - 2c_{3} = -9 \). Solve the new 2x2 system: \[ \begin{align*} 3c_{2} + 2c_{3} &= 5 \ c_{2} - 2c_{3} &= -9 \end{align*} \] Adding equations to eliminate \( c_3 \):\( 4c_{2} = -4 \Rightarrow c_{2} = -1 \). Substituting \( c_{2} = -1 \) back:\( -1 - 2c_{3} = -9 \Rightarrow 2c_{3} = 8 \Rightarrow c_{3} = 4 \). Finally, \( c_{1} = 6 - 4 = 2 \). The values satisfy the equations confirming a solution.

Conclusion for Case b

The result \( c_1 = 2, c_2 = -1, c_3 = 4 \) and that the system of equations has consistent solutions means \( \mathbf{y} \) is a linear combination of \( \mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3} \) in Case b.

Key concepts

Vector Spaces

Vector spaces are fundamental structures in linear algebra. Imagine a set of vectors, where you can perform operations like vector addition and scalar multiplication, and the results will still be another vector in the same set. This collection of vectors and these operations together form a 'vector space'.

- **Vector Addition**: This is combining two vectors to produce a third. For example, if you have vectors \( extbf{u} \) and \( extbf{v} \), their sum is \( extbf{u} + extbf{v} \).- **Scalar Multiplication**: Here, you multiply a vector by a scalar (a number), resulting in a scaled version of the original vector. \(c extbf{u} \) scales \( extbf{u} \) by \( c \).
These operations must satisfy certain rules, such as associativity and distributivity, to qualify the set as a vector space. Whether in physics, engineering, or computer science, understanding vector spaces aids in simplifying complex systems by breaking them down into manageable vectors and operations.

System of Linear Equations

Systems of linear equations are equations where each one is a linear equation. They enable us to determine the values of unknowns (variables) that satisfy all these linear conditions simultaneously. A system of equations is typically solved using methods like substitution, elimination, or matrix row operations.

- **Consistency**: A system is consistent if there exists at least one solution to all the equations together.- **Inconsistency**: No common solution satisfies all equations simultaneously.
For our exercise, when checking if a vector \(\textbf{y} \) can be expressed as a linear combination of vectors \( \textbf{a}_1, \textbf{a}_2, \textbf{a}_3 \), we essentially form and solve a system of equations. This lets us see if there exist scalars \( c_1, c_2, c_3 \) that make \( \textbf{y} = c_1 \textbf{a}_1 + c_2 \textbf{a}_2 + c_3 \textbf{a}_3 \) true.

Linear Algebra

Linear Algebra is the branch of mathematics concerning linear equations and their representations through matrices and vector spaces. It's essentially a language for expressing mathematical concepts in dimensions of space.

- **Linear Combinations**: Involves constructing new vectors using a set of vectors, multi-scaled and added together. This foundational concept is pivotal for expressing solutions in various computations. - **Matrices**: Arrays of numbers used to organize and solve systems of linear equations. A matrix can help represent and solve complex systems involving many unknowns.
Understanding linear algebra is crucial to solving problems in physics, engineering, and computer graphics, structuring the problem in a way that makes its solution more accessible and systematic.

Computer Graphics Mathematics

Computer graphics involve rendering images and animations, requiring heavy use of mathematics, especially linear algebra.

- **Vectors and their Transformations**: Objects in graphics are represented and manipulated as vectors. Linear combinations let artists and computers transform objects by rotating, scaling, and translating them in graphic scenes. - **Matrices in Graphics**: Use matrices to effect transformations. For instance, using a matrix, one can rotate an image or change its scale by applying simple multiplication to its corresponding vectors.
Through linear combinations, one effectively performs these transformations, crucial for visual effects, game engines, and virtual reality applications. Mastery of these mathematical concepts empowers developers and artists to bring complex three-dimensional animations to life.