A question about linear transformation (One-One /Onto)

Question

Suppose $T:\mathbb{R}^4 \to \mathbb{R}^4$ is the transformation given below . Determine whether $T$ is one-to-one and/or onto. if it is not one-to-one . Show this by providing tow vectors that have the same image under $T$. I T is not onto. Show this by providing a vector in $\mathbb{R}^4$ that is not in the range of $$

$$T\begin{bmatrix} x_0\\ x_1\\ x_2\\ x_3 \end{bmatrix}=\begin{bmatrix} 2x_0-10x_1-4x_2-2x_3\\ x_0-5x_1+x_2+5x_3\\ -x_0+5x_1+3x_2+4x_3\\ -2x_0+10x_1+3x_2+x_3 \end{bmatrix}$$

My attempt : the matrix form standard basis is

$$\begin{bmatrix} 2 &-10 &-10 &-2 \\ 1 &-5 &1 &5 \\ -1&5 &3 &4 \\ -2&10 &3 &1 \end{bmatrix}\sim\begin{bmatrix} 1 &-5 &1 &5 \\0 &0 &1 &2 \\0 &0 &0 &1 \\0 &0 &0 &0 \end{bmatrix}$$

So this transformation is neither one-to-one or onto

but i cant find examples .Is there any way to find simple way to give counter examples to show for not one-to-one or not onto

Answer 1

We make the following augmented system $$ \begin{bmatrix} 2&-10&-4&-2&a\\ 1&-5&1&5&b\\ -1&5&3&4&c\\ -2&10&3&1&d \end{bmatrix} $$ inconsistent, and so find a vector not in the image. By row reducing (barring any mistakes) we get a row of zeros equal to the following $$ a+6d+4b-6c $$ as long as we insure this is nonzero, we have an inconsistent system. One may check that, for example, the vector $$ \begin{bmatrix}1\\0\\0\\0 \end{bmatrix} $$ is not in the range.

Answer 2

$\begin{pmatrix}1 & -5 & 1 & 5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{pmatrix}\begin{pmatrix}a \\ b\\c\\d\end{pmatrix} = 0$

From the third row we see $d=0$ , from the second row combined with the knowledge that $d=0$ we see that $c+2d = c+0 = 0$, so $c=0$ as well.

So overall we have now from the first row $a-5b = 0$, or $a=5b$.

So every vector you choose that looks like $\begin{pmatrix}5t \\ t \\0 \\0\end{pmatrix}$ will go to $0$, so this mapping isn't one to one.