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Problems and solutions in Linear Algebra


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  • Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis.Three Linearly Independent Vectors in $\R^3$ Form a Basis. Three Vectors Spanning $\R^3$ Form a Basis. Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$. (a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$. (b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of […]
  • Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent? Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$. Consider the functions \[f(x)=\sin^2(x) \text{ and } g(x)=\cos^2(x)\] in $C[-2\pi, 2\pi]$. Prove or disprove that the functions $f(x)$ and $g(x)$ are linearly […]
  • Determine Trigonometric Functions with Given ConditionsDetermine Trigonometric Functions with Given Conditions (a) Find a function \[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\] such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants. (b) Find real numbers $a, b, c$ such that the function \[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 […]
  • Find the Largest Prime Number Less than One Million.Find the Largest Prime Number Less than One Million. Find the largest prime number less than one million. What is a prime number? A natural number is called a "prime number" if it is only divisible by $1$ and itself. For example, $2, 3, 5, 7$ are prime numbers, although the numbers $4,6,9$ are not. The prime numbers have always […]
  • Equivalent Definitions of Characteristic Subgroups. Center is Characteristic.Equivalent Definitions of Characteristic Subgroups. Center is Characteristic. Let $H$ be a subgroup of a group $G$. We call $H$ characteristic in $G$ if for any automorphism $\sigma\in \Aut(G)$ of $G$, we have $\sigma(H)=H$. (a) Prove that if $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$, then $H$ is characteristic in $G$. (b) Prove that the center […]
  • Group of Order $pq$ Has a Normal Sylow Subgroup and SolvableGroup of Order $pq$ Has a Normal Sylow Subgroup and Solvable Let $p, q$ be prime numbers such that $p>q$. If a group $G$ has order $pq$, then show the followings. (a) The group $G$ has a normal Sylow $p$-subgroup. (b) The group $G$ is solvable.   Definition/Hint For (a), apply Sylow's theorem. To review Sylow's theorem, […]
  • Determinant/Trace and Eigenvalues of a MatrixDeterminant/Trace and Eigenvalues of a Matrix Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues. Show that (1) $$\det(A)=\prod_{i=1}^n \lambda_i$$ (2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$ Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix […]
  • Determine a Value of Linear Transformation From $\R^3$ to $\R^2$Determine a Value of Linear Transformation From $\R^3$ to $\R^2$ Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that \[ T\left(\, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\,\right) =\begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{ and }T\left(\, \begin{bmatrix} 0 \\ 1 \\ 1 […]

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