Question

Question 7 Determine whether the linear transformation T is one-to-one and whether it maps as specified. 2 + 3x 3) T(X 1, X 2, x 3) = (-2x 2 - 2x 3, -2x 1 + 8x 2 + 4x 3, -X 1 - 2x 3,3x Determine whether the linear transformation T is one-to-one and whether it maps R 3 onto R4. Not one-to-one; not onto R4 One-to-one; onto R4 Not one-to-one; onto R4 One-to-one; not onto R4

Answer 1

for a transformation to be one to one , we look whether the columns of the transformation matrix is independent by converting the matrix to ro reduced echelon form , if the columns are independent then the transformation is one to one here we get the matrix of transformation as

$T(x)=\begin{bmatrix} 0 &-2 & -2\\ -2& 8& 4\\ -1& 0 & -2\\ 0 & 3 & 3 \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}$

the matrix is

-2 8 -2 4 A = 0 -2 -1 ] -2 3 3

$R_2 \leftrightarrow R_1$

$A=\begin{bmatrix} -2 &8 & 4\\ 0& -2& -2\\ -1& 0 & -2\\ 0 & 3 & 3 \end{bmatrix}$

$R_2 \to R_2/-2$

$A=\begin{bmatrix} 1 &-4 & -2\\ 0& -2& -2\\ -1& 0 & -2\\ 0 & 3 & 3 \end{bmatrix}$

$R_3 \to R_3+R_1$

$A=\begin{bmatrix} 1 &-4 & -2\\ 0& -2& -2\\ 0& -4 & -4\\ 0 & 3 & 3 \end{bmatrix}$

$R_2\to R_2/-2$

$A=\begin{bmatrix} 1 &-4 & -2\\ 0& 1& 1\\ 0& -4 & -4\\ 0 & 3 & 3 \end{bmatrix}$

$R_1 \to R_1+4R_2$

$R_3 \to R_3+4R_2$

$R_4 \to R_4-3R_2$

$A=\begin{bmatrix} 1 &0 & 2\\ 0& 1& 1\\ 0& 0 & 0\\ 0 & 0 & 0 \end{bmatrix}$

so the row reduced form of A does not any pivots in columns or rows ,

So the transformation is not one to one and not onto