(a) Write down an example of a system of 5 linear equations...

(a) Write down an example of a system of 5 linear equations in 5 unknowns with regular diagonal coefficient matrix. (b) Solve your system. (c) Explain why solving a system whose coefficient matrix is diagonal is very easy. 1.3.7. Find the equation of the parabola $y=a x^2+b x+c$ that goes through the points $(1,6),(2,4)$, and $(3,0)$. 1.3.8. A linear system is called homogeneous if all the right-hand sides are zero, and so takes the matrix form $A \mathbf{x}=\mathbf{0}$. Explain why the solution to a homogeneous system with regular coefficient matrix is $\mathbf{x}=\mathbf{0}$.

(a) Write down an example of a system of 5 linear equations in 5 unknowns with regular diagonal coefficient matrix. (b) Solve your system. (c) Explain why solving a system whose coefficient matrix is diagonal is very easy.
1.3.7. Find the equation of the parabola $y=a x^2+b x+c$ that goes through the points $(1,6),(2,4)$, and $(3,0)$.
1.3.8. A linear system is called homogeneous if all the right-hand sides are zero, and so takes the matrix form $A \mathbf{x}=\mathbf{0}$. Explain why the solution to a homogeneous system with regular coefficient matrix is $\mathbf{x}=\mathbf{0}$.

Key Concepts

System of Linear Equations

A system of linear equations consists of several equations involving the same set of unknowns. The solution to the system is the set of values for the unknowns that satisfy all equations simultaneously. Such systems can be compactly expressed in matrix form, allowing the use of linear algebra techniques for analysis and solution.

Regular (Invertible) Matrix

A regular matrix, also known as an invertible or nonsingular matrix, is a square matrix that has a nonzero determinant. This property guarantees the existence of a unique solution for the matrix equation Ax = b, making it fundamental in solving systems of linear equations using methods such as matrix inversion.

Diagonal Coefficient Matrix

A diagonal coefficient matrix is a special type of square matrix in which all off-diagonal entries are zero. This structure simplifies the process of solving the associated linear system because each equation becomes independent of the others, allowing for the straightforward computation of each unknown by directly dividing the corresponding constant term by the diagonal element.

Quadratic Functions (Parabolas)

A quadratic function is a second-degree polynomial function, typically written in the form y = ax^2 + bx + c. The graph of this function is a parabola. Finding a quadratic function that passes through given points involves setting up and solving a system of linear equations to determine the coefficients a, b, and c, which uniquely define the parabola.

Homogeneous Systems

A homogeneous system of linear equations is one in which all of the constant terms (the right-hand sides) are zero, taking the form Ax = 0. When the coefficient matrix A is regular or invertible, the only solution is the trivial solution (x = 0) because any nonzero vector would contradict the invertibility property, which ensures that the system has a unique solution.

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