Question

If T:ℝ3→ℝ3 is a linear transformation such that T⎛⎝⎜⎜⎡⎣⎢⎢100⎤⎦⎥⎥⎞⎠⎟⎟=⎡⎣⎢⎢3−2−3⎤⎦⎥⎥, T⎛⎝⎜⎜⎡⎣⎢⎢010⎤⎦⎥⎥⎞⎠⎟⎟=⎡⎣⎢⎢401⎤⎦⎥⎥, T⎛⎝⎜⎜⎡⎣⎢⎢001⎤⎦⎥⎥⎞⎠⎟⎟=⎡⎣⎢⎢−1−24⎤⎦⎥⎥, then T⎛⎝⎜⎜⎡⎣⎢⎢3−4−2⎤⎦⎥⎥⎞⎠⎟⎟= ⎡⎣⎢⎢⎢⎢⎢⎢ ⎤⎦⎥⎥⎥⎥⎥⎥

Answer 1

In this case, the value of T is
`T=\left[\begin{array}{ccc} 2 \\ 5 \\0\end{array}\right]`

Step-by-step explanation:

To find the value of T, we can use the given information about the linear transformation T:R³ -> R³ and the transformation of the standard basis vectors.

We are given:

`T\left[\begin{array}{ccc}1 \\0\\0\end{array}\right] =\left[\begin{array}{ccc}-4\\-4\\1\end{array}\right] T\left[\begin{array}{ccc}0\\1\\0\end{array}\right] =\left[\begin{array}{ccc}-3\\0\\1\end{array}\right] T\left[\begin{array}{ccc}0\\0\\1\end{array}\right] =\left[\begin{array}{ccc}-1 \\-4\\3\end{array}\right]`

We can write the transformation matrix for T using the images of the standard basis vectors as columns:

`T= \left[\begin{array}{ccc}T([1, 0, 0]) \\T([0, 1, 0])\\T([0, 0, 1])\end{array}\right]= \left[\begin{array}{ccc}(-4, -3, -1)\\(-4, 0, -4)\\(1, 1, 3)\end{array}\right]`

To find the value of T, we multiply the transformation matrix by the given vector:

`T*\left[\begin{array}{ccc}-1\\-4\\3\end{array}\right]= \left[\begin{array}{ccc}(-4)(-1) + (-3)(-4) + (-1)(3)\\(-4)(-4) + 0 + (-4)(3)\\ (1)(-1) + 1 + (3)(3)\end{array}\right]= \left[\begin{array}{ccc} 2 \\ 5 \\0\end{array}\right]`

Therefore,
`T=\left[\begin{array}{ccc} 2 \\ 5 \\0\end{array}\right]`.

The correct answer is
`T=\left[\begin{array}{ccc} 2 \\ 5 \\0\end{array}\right]`.

Your question is incomplete, but most probably the full question was:

If T:R³ →R³ is a linear transformation such that

`T \left[\begin{array}{ccc}1 \\0 \\0\end{array}\right] = \left[\begin{array}{ccc}-4\\-4\\1\end{array}\right] , T\left[\begin{array}{ccc}0\\1\\0\end{array}\right] =\left[\begin{array}{ccc}-3\\0\\1\end{array}\right] ,T=\left[\begin{array}{ccc}0\\0\\1\end{array}\right] \left[\begin{array}{ccc}-1\\-4\\3\end{array}\right] ,then \T=\left[\begin{array}{ccc}-2\\ 5 \\0\end{array}\right]=?`