Find out whether the given sets of matrices are linearly dependent in $M_{22}$.

(a)

$$\left\{\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 2 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 3 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 4\end{array}\right]\right\}$$

(b)

$$\left\{\left[\begin{array}{ll}1 & 2 \\ 3 & 1\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}2 & 1 \\ 4 & 2\end{array}\right]\right\}$$

(c)

$$\left\{\left[\begin{array}{rr}1 & 2 \\ -1 & 0\end{array}\right],\left[\begin{array}{ll}1 & 2 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 2 \\ 5 & 3\end{array}\right]\right\}$$

(d)

$$\left\{\left[\begin{array}{ll}2 & 4 \\ 0 & 1\end{array}\right],\left[\begin{array}{rr}0 & -2 \\ 8 & 3\end{array}\right],\left[\begin{array}{rr}-3 & -7 \\ 4 & 0\end{array}\right]\right\}$$

Is the transformation $T(a x+b)=a x^2+a x+a$ of $P_1 \rightarrow P_2$ linear? Determine the image of $2 x+1$. Find another element of $P_1$ that has the same image.

The given sets are all linearly independent. Add a vector to each set that will make the new sets linearly dependent.

(a) $\{(1,-1,0),(2,1,3)\}$ (b) $\{(3,-5,1),(1,4,3)\}$ (c) $\{(1,2,4),(0,2,5)\}$ (d) $\{(3,-2,4),(2,5,7)\}$