Dr. Xiaozhou Li

Dr. Xiaozhou Li

Mathematics. Strength. Long-Term Mindset. Think Hard. Train Harder. Live Long.

Problem Sets: Systems of Linear Equations (线性方程组)

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Last Update: 2026-03-31

  • In general, what is the elementary row operation that “undoes” each of the three elementary row operations $R_i \leftrightarrow R_j$, $kR_i$, and $R_i + kR_j$?

  • For what values of $a, b$ the homogeneous linear system \(\begin{cases} ax_1 + bx_2 + bx_3 + \cdots + bx_n = 0, \\ bx_1 + ax_2 + bx_3 + \cdots + bx_n = 0, \\ \cdots\cdots \\ bx_1 + bx_2 + bx_3 + \dots + ax_n = 0 \end{cases}\) has only the trivial solution, infinitely many solutions. When there are infinitely many solutions, find all solutions.
  • Given the set of vectors \(\vec{\alpha}_1 = \begin{bmatrix} 1+\lambda \\ 1 \\ 1 \end{bmatrix}, \quad \vec{\alpha}_2 = \begin{bmatrix} 1 \\ 1+\lambda \\ 1 \end{bmatrix}, \quad \vec{\alpha}_3 = \begin{bmatrix} 1 \\ 1 \\ 1+\lambda \end{bmatrix}, \quad \vec{\beta} = \begin{bmatrix} 0 \\ \lambda \\ \lambda^2 \end{bmatrix},\)for what values of $\lambda$:
    1. $\vec{\beta}$ can be written as an unique linear combination of $\vec{\alpha}_1, \vec{\alpha}_2, \vec{\alpha}_3$?
    2. $\vec{\beta}$ can be expressed as many different linear combination of $\vec{\alpha}_1, \vec{\alpha}_2, \vec{\alpha}_3$?
    3. $\vec{\beta}$ can not be written as a linear combination of $\vec{\alpha}_1, \vec{\alpha}_2, \vec{\alpha}_3$?

  • For what values of $a$ and $b$ does the system of linear equations \(\begin{cases} x_1 + x_2 + x_3 + x_4 = 0, \\ x_2 + 2x_3 + 2x_4 = 1, \\ -x_2 + (a-3)x_3 - 2x_4 = b, \\ 3x_1 + 2x_2 + x_3 + ax_4 = -1 \end{cases}\) have a unique solution, no solution, or infinitely many solutions? When the system is consistent, find the solution.

  • Consider the following two nonhomogeneous linear systems (I) and (II): \(\text{(I): } \begin{cases} x_1 + x_2 - 2x_4 = -6, \\ 4x_1 - x_2 - x_3 - x_4 = 1, \\ 3x_1 - x_2 - x_3 = 3, \end{cases}\)and \(\text{(II): } \begin{cases} x_1 + mx_2 - x_3 - x_4 = -5, \\ nx_2 - x_3 - 2x_4 = -11, \\ x_3 - 2x_4 = -t + 1. \end{cases}\)
    1. Solve system (I).
    2. Find the values of the parameters $m, n, t$ in system (II) such that systems (I) and (II) are equivalent.
  • Let the homogeneous linear system (I) be \(\text{(I): } \begin{cases} x_1 + x_2 = 0, \\ x_2 - x_4 = 0. \end{cases}\)It is known that the general solution of another homogeneous linear system (II) is \(c_1[0,1,1,0]^{T} + c_2[-1,2,2,1]^{T}.\)
    1. Find the general solution for linear system (I).
    2. Determine whether linear systems (I) and (II) have any non-trivial common solutions. If yes, find all such solutions.

  • Find a $2 \times 3$ system (2 equations with 3 unknowns) such that its general solution has a form \(\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + s \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad s \in \mathbb{R}.\)

  • Given vectors \(\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \quad \begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix}\)Do these four vectors span $\mathbb{R}^4$?

  • Are the matrices \(\begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & -1 \\ -1 & 4 & 1 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} 1 & 0 & -1 \\ 1 & 1 & 1 \\ 0 & 1 & 3 \end{bmatrix}\) row equivalent? Why or why not?

  • Solve \(\begin{cases} x_1 + 5x_2 - x_3 - x_4 = -1, \\ x_1 - 2x_2 + x_3 + 3x_4 = 3, \\ 3x_1 + 8x_2 - x_3 + x_4 = 1, \\ x_1 - 9x_2 + 3x_3 + 7x_4 = 7. \end{cases}\)

  • Solve \(\begin{cases} x_1 + x_2 + x_3 + x_4 = 1, \\ x_1 + 2x_2 + 2x_3 + 2x_4 = 0, \\ x_1 + 2x_2 + 3x_3 + 3x_4 = 0, \\ x_1 + 2x_2 + 3x_3 + 4x_4 = 0. \end{cases}\)

  • Let $\vec{\alpha}_1 = [1, 0, 1, 0]^{T}$, $\vec{\alpha}_2 = [2, 2, a, 2]^{T}$, $\vec{\alpha}_3 = [3, 1, 1, 1]^{T}$, $\vec{\beta} = [4, -1, 6, b]^{T}$. Find $a, b$ such that $\vec{\beta}$ is not a linear combination of $\vec{\alpha}_1,\vec{\alpha}_2,\vec{\alpha}_3$, and find $a,b$ such that it is. Write the linear combination when possible.

  • Let $\vec{u}_1 = [-9, 1, 2, 11]^{T}$, $\vec{u}_2 = [1, -5, 13, 0]^{T}$, $\vec{u}_3 = [-7, -9, 24, 11]^{T}$ be three solutions to the linear system \(\begin{cases} 2x_1 + a_2 x_2 + 3x_3 + a_4 x_4 = d_1, \\ 3x_1 + b_2 x_2 + 2x_3 + b_4 x_4 = 4, \\ 9x_1 + 4x_2 + x_3 + c_4 x_4 = d_3. \end{cases}\)Determine its general solution.

  • Let $A$ be an $n \times n$ real matrix. Show that $A\vec{x} = \vec{0}$ and $A^{T} A\vec{x} = \vec{0}$ are equivalent linear systems.

  • By solving a $3 \times 3$ system, find the coefficients in the equation of the parabola $y = \alpha + \beta x + \gamma x^{2}$ that passes through the points $(1, 1)$, $(2, 2)$, and $(3, 0)$.

  • Suppose that $A$ is the coefficient matrix for a homogeneous system of four equations in six unknowns and suppose that $A$ has at least one nonzero row.
    1. Determine the fewest number of free variables that are possible.
    2. Determine the maximum number of free variables that are possible.

  • Explain why a homogeneous system of $m$ equations in $n$ unknowns where $m < n$ must always possess an infinite number of solutions.

  • Construct a homogeneous system of three equations in four unknowns that has \(x_2 \begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \end{bmatrix} + x_4 \begin{bmatrix} -3 \\ 0 \\ 2 \\ 1 \end{bmatrix}\) as its general solution.

  • Among the solutions that satisfy the set of linear equations \(\begin{cases} x_1 + x_2 + 2x_3 + 2x_4 + x_5 = 1, \\ 2x_1 + 2x_2 + 4x_3 + 4x_4 + 3x_5 = 1, \\ 2x_1 + 2x_2 + 4x_3 + 4x_4 + 2x_5 = 2, \\ 3x_1 + 5x_2 + 8x_3 + 6x_4 + 5x_5 = 3, \end{cases}\) find all those that also satisfy the following two constraints: \(\begin{cases} (x_1 - x_2)^{2} - 4x_5^{2} = 0, \\ x_3^{2} - x_5^{2} = 0. \end{cases}\)