# Linear algebra questions and answers

Recent questions in Linear algebra
Liberty Mack 2022-05-20 Answered

### Finding alternate transformation matrix for similarity transformationA pair of square matrices $X$ and $Y$ are called similar if there exists a nonsingular matrix $T$ such that ${T}^{-1}XT=Y$ holds. It is known that the transformation matrix $T$ is not unique for given $X$ and $Y$. I'm just wondering whether those non-unique transformation matrices would have any relation among themselves, like having column vectors with same directions...What I want to mean is: Given $X$ and $Y$ a pair of similar matrices, if $S$ and $T$ are two possible transformation matrices satisfying ${S}^{-1}XS={T}^{-1}XT=Y$, is there any generic (apart from scaling) relation between $T$ and $S$ (e.g., direction of column vectors)?For a specific example, consider $X=\left[\begin{array}{cc}A& BK\\ C& 0\end{array}\right]$ and $Y=\left[\begin{array}{cc}A+{A}^{-1}BKC& -{A}^{-1}BKC{A}^{-1}B\\ KC& -KC{A}^{-1}B\end{array}\right]$. Assuming $K$ to be invertible it can be shown that $X$ and $Y$ are similar with transformation matrix $T=\left[\begin{array}{cc}I& -{A}^{-1}B\\ 0& {K}^{-1}\end{array}\right]$. Can there be any other matrix $S$ which will be independent of $K$, and would result ${S}^{-1}XS=Y$?

Jazmine Bruce 2022-05-19 Answered

### What is the general transformation matrix for a rotation of angle $\theta$ about the origin?That is all the questions says, any one able to help me out, who may understand it better then me?

Camille Flynn 2022-05-19 Answered

### The form $f\left(x\right){y}^{n}\left(x\right)+...u\left(x\right)y\left(x\right)=h\left(x\right)$ is supposed to be the definition of lineaity in diff equations. It excludes functions of y in the right hand side, but is multiplication of y by another function allowed in the right hand side allowed? It seems to be the case sometimes but not all of the time, as only composition of linear functions gives linear functions. Can't we just move it to the other side with the other y and call that form linear as well?

Chaz Blair 2022-05-19 Answered

### Let $f:{\mathbb{R}}^{m}\to {\mathbb{R}}^{n}$ be a linear transformation. As is common knowledge, it can be expressed as an $n×m$ matrix.

Rachel Villa 2022-05-18 Answered

### I am trying to show that the functions ${t}^{3}$ and $b$ are independent on the whole real line. To do this, I try and prove it by contradiction. So assume that they are dependent. So then there must exists constants $a$ and $b$ such that $a{t}^{3}+b|t{|}^{3}=0$ for all $t\in \left(-\mathrm{\infty },\mathrm{\infty }\right)$. Now pick two points $x$ and $y$ in this interval and assume without loss of generality that $x<0$, $y\ge 0$. Now form the simultaneous linear equations$a{y}^{3}+b|y{|}^{3}=0$, viz.$\left[\begin{array}{cc}{x}^{3}& |x{|}^{3}\\ {y}^{3}& |y{|}^{3}\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$Now here's my problem. If I look at the determinant of the coefficient matrix of this system of linear equations, namely ${x}^{3}|y{|}^{3}-{y}^{3}|x{|}^{3}$ and noting that $x<0$ and $y>0$, I have that the determinant is non-zero which implies that the only solution is $a=b=0$, i.e. the functions ${t}^{3}$ and |$|t{|}^{3}$ are linearly independent. However what happens if indeed $y=0$? Then the determinant of the matrix is 0 and I have got a problem.Is there something that I am not getting from the definition of linear independence?The definition (I hope I state this correctly) is: If $f$ and $g$ are two functions such that the only solution to in an interval $I$ is $a=b=0$, then the two functions are linearly independent.But what happens if my functions pass through the origin, like the above? Then I've just shown that there exists a $t$ in an interval containing zero such that the two functions are zero, viz. I can plug in any $b$ and $a$ such that $af+bg=0$.

Cesar Mcguire 2022-05-17 Answered

### Let $T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{3}$ and $T\left(-2,3\right)=\left(-1,0,1\right)$ and $T\left(1,2\right)=\left(0,-1,0\right)$Obtain the canonical matrix of $T$ and the transformation $T\left(x,y\right)$.

Jaylene Duarte 2022-05-15 Answered