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• Determine Conditions on Scalars so that the Set of Vectors is Linearly Dependent Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent. \begin{align*} S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}, \end{align*} where \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, […]
• True or False Quiz About a System of Linear Equations (Purdue University Linear Algebra Exam)   Which of the following statements are true? (a) A linear system of four equations in three unknowns is always inconsistent. (b) A linear system with fewer equations than unknowns must have infinitely many solutions. (c) […]
• Using Properties of Inverse Matrices, Simplify the Expression Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression \[C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,\] which matrix do you get? (a) $A$ (b) $C^{-1}A^{-1}BC^{-1}AC^2$ (c) $B$ (d) $C^2$ (e) $C^{-1}BC$ (f) $C$   Solution. In this problem, we […]
• Linear Transformation $T:\R^2 \to \R^2$ Given in Figure Let $T:\R^2\to \R^2$ be a linear transformation such that it maps the vectors $\mathbf{v}_1, \mathbf{v}_2$ as indicated in the figure below. Find the matrix representation $A$ of the linear transformation $T$.   Solution 1. From the figure, we see […]
• Quiz 2. The Vector Form For the General Solution / Transpose Matrices. Math 2568 Spring 2017. (a) The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. \[ \left[\begin{array}{rrrrr|r} 1 & 0 & -1 & 0 &-2 & 0 \\ 0 & 1 & 2 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ \end{array} \right].\] […]
• Differentiating Linear Transformation is Nilpotent Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less. Consider the differentiation linear transformation $T: P_n\to P_n$ defined by \[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\] (a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a […]
• Quiz 1. Gauss-Jordan Elimination / Homogeneous System. Math 2568 Spring 2017. (a) Solve the following system by transforming the augmented matrix to reduced echelon form (Gauss-Jordan elimination). Indicate the elementary row operations you performed. […]
• Image of a Normal Subgroup Under a Surjective Homomorphism is a Normal Subgroup Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$. Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.   Proof. To show that $f(N)$ is normal, we show that $gf(N)g^{-1}=f(N)$ for any $g \in […]