(10%) In R^2, there are two sets of coordinate systems, represented by two distinct bases: (x_1, y_1) and (x_2, y_2). If the equations of the same ellipse represented

Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has an infinite number of solutions.

Find a basis for the span of the given vectors.
[0 1 -2 1], [3 1 -1 0], [2 1 5 1]

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 $t\in (-\mathrm{\infty },\mathrm{\infty })$. 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 $af+bg=0\mathrm{\forall }t$ 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 $af+bg=0$.

(P^~Q)^(P→Q) 

check whether it is in contradiction form or not.

List five vectors in Span ${\displaystyle \{v_1,v_2\}}$. For each vector, show the weights on ${\displaystyle v_1}$ and ${\displaystyle v_2}$ used to generate the vector and list three entries of the vector. Do not make a seketch.
$v_1=\left[\begin{array}{c}7\\ 1\\ -6\end{array}\right],v_2=\left[\begin{array}{c}-5\\ 3\\ 0\end{array}\right]$

Transformation Matrix of a dot product transformation
Let v be a arbitrary vector in R3 and T(x)=v.x. What is the matrix of the transformation T in terms of the components of v? It seems like trying to figure out the matrix using the equation T(x)=Ax does not work, as the left side is a scalar, and the other side is a matrix. Any idea

Need to find a matrix transformation that takes a (non-square) matrix and within each of its rows keeps the first nonzero element in that row and zeros out the rest of the entries within that row.
I tried to solve the linear matrix equation AX = B to get the matrix transformation A as A = B X^T (XX^T)^{-1} , where X is the matrix to be transformed as such, and B is the desired output matrix (X with only its first nonzero element in each row) but I look for a more straightforward and possibly more elegant way to do it. I appreciate creative answers!