# system-of-linear-equations-eye-catch

by Yu · Published · Updated

Add to solve later

Add to solve later

Add to solve later

### More from my site

- Group of Order 18 is Solvable Let $G$ be a finite group of order $18$. Show that the group $G$ is solvable. Definition Recall that a group $G$ is said to be solvable if $G$ has a subnormal series \[\{e\}=G_0 \triangleleft G_1 \triangleleft G_2 \triangleleft \cdots \triangleleft G_n=G\] such […]
- Linearly Dependent if and only if a Vector Can be Written as a Linear Combination of Remaining Vectors Let $V$ be a vector space over a scalar field $K$. Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ be the set of vectors in $V$, where $n \geq 2$. Then prove that the set $S$ is linearly dependent if and only if at least one of the vectors in $S$ can be written as […]
- Determine Whether Given Subsets in $\R^4$ are Subspaces or Not (a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying \[2x+4y+3z+7w+1=0.\] Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a […]
- Determine Whether Trigonometry Functions $\sin^2(x), \cos^2(x), 1$ are Linearly Independent or Dependent Let $f(x)=\sin^2(x)$, $g(x)=\cos^2(x)$, and $h(x)=1$. These are vectors in $C[-1, 1]$. Determine whether the set $\{f(x), \, g(x), \, h(x)\}$ is linearly dependent or linearly independent. (The Ohio State University, Linear Algebra Midterm Exam […]
- Find All Values of $a$ which Will Guarantee that $A$ Has Eigenvalues 0, 3, and -3. Let $A$ be the matrix given by \[ A= \begin{bmatrix} -2 & 0 & 1 \\ -5 & 3 & a \\ 4 & -2 & -1 \end{bmatrix} \] for some variable $a$. Find all values of $a$ which will guarantee that $A$ has eigenvalues $0$, $3$, and $-3$. Solution. Let $p(t)$ be the […]
- Is the Determinant of a Matrix Additive? Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$. Is it true that \[\det(A+B)=\det(A)+\det(B)?\] If so, then give a proof. If not, then give a counterexample. Solution. We claim that the statement is false. As a counterexample, […]
- A Basis for the Vector Space of Polynomials of Degree Two or Less and Coordinate Vectors Show that the set \[S=\{1, 1-x, 3+4x+x^2\}\] is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less. Proof. We know that the set $B=\{1, x, x^2\}$ is a basis for the vector space $P_2$. With respect to this basis $B$, the coordinate […]
- Find Matrix Representation of Linear Transformation From $\R^2$ to $\R^2$ Let $T: \R^2 \to \R^2$ be a linear transformation such that \[T\left(\, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 4 \\ 1 \end{bmatrix}, T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \,\right)=\begin{bmatrix} 3 \\ 2 […]