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Matrix$\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$is similar to$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$.

In Exercises 10through 20, use paper and pencil to identify the redundant vectors. Thus determine whether the given vectors are linearly independent.

15.$\left[\begin{array}{c}1\\ 2\end{array}\right]\mathbf{,}\left[\begin{array}{c}2\\ 3\end{array}\right]\mathbf{,}\left[\begin{array}{c}3\\ 4\end{array}\right]$

Vectors $\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\mathit{}\left[\begin{array}{c}2\\ 1\\ 3\end{array}\right],\mathit{}\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right]$ form a basis of$R^3$ .

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.

16.$\left[\begin{array}{ccccc}1& -2& 0& -1& 0\\ 0& 0& 1& 5& 0\\ 0& 0& 0& 0& 1\end{array}\right]$

In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.

17. $\left[\begin{array}{ccccc}0& 1& 2& 0& 3\\ 0& 0& 0& 1& 4\end{array}\right]$

In Exercises 10through 20, use paper and pencil to identify the redundant vectors. Thus determine whether the given vectors are linearly independent.

$15.\left[\begin{array}{c}1\\ 2\end{array}\right],\left[\begin{array}{c}2\\ 3\end{array}\right],\left[\begin{array}{c}3\\ 4\end{array}\right]$

If the kernel of a matrix A consists of zero vector only, then the column vectors of A must be linearly independent.

In Exercises 10through 20, use paper and pencil to identify the redundant vectors. Thus determine whether the given vectors are linearly independent.

18.$\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\mathbf{,}\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\mathbf{,}\left[\begin{array}{c}3\\ 0\\ 0\end{array}\right]\mathbf{,}\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]\mathbf{,}\left[\begin{array}{c}4\\ 5\\ 0\end{array}\right]\mathbf{,}\left[\begin{array}{c}6\\ 7\\ 0\end{array}\right]\mathbf{,}\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$.

If the image of an n x n matrix A is all of $R^n$, then A must be invertible.

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.

18.$$\begin{gathered} \left[1151 \\ 0122 \\ 0123 \\ 0124\right] \end{gathered}$$