Question

Let t : R³ → R³ be a linear transformation such that t(1, -1, 1) = (3, 1, 2), t(0, 1, 0) = (5, ?, ?), and t(4, ?, ?) = (2, ?, ?). Find the values of t(4, ?, ?).

Answer 1

To find the values of t(4, ?, ?), substitute the values into the equation for the linear transformation t. The values are (12, 4-?, 8+?).

Step-by-step explanation:

To find the values of t(4, ?, ?), we need to use the given information about the linear transformation t. We know that t(1, -1, 1) = (3, 1, 2), which gives us the equation:

t(x, y, z) = (3x, x-y, 2x+y).

Now, we can use this equation to find t(0, 1, 0) and t(4, ?, ?).

For t(0, 1, 0), we substitute the values into the equation:

t(0, 1, 0) = (3(0), 0-1, 2(0)+1) = (0, -1, 1).

Similarly, for t(4, ?, ?), we substitute the values into the equation:

t(4, ?, ?) = (3(4), 4-?, 2(4)+?) = (12, 4-?, 8+?).

So, the values of t(4, ?, ?) are (12, 4-?, 8+?).