Properties of Linear Transformations from R^n to R^m?

Question

1) Find the standard matrix for the linear operator defined by the equations, determine whether the operator is one-to-one.
w_1 = 2x_1 - 3x_2
w_2 = 5x_1 + x_2

2) Show that the range of the linear operator defined by the equations
w_1 = x_1 - 2x_2 + x_3
w_2 = 5x_1 - x_2 +3x_3
w_3 = 4x_1 + x_2 + 2x_3
is not all of R^3, and find a vector that is not in the range.

Answer 1

1) This tx is W =
[2 -3] X
[5 1],
where W is the column vector [w_1 w_2}^T and X = [x_1 x_2]^T. It is 1 - 1 because the matrix is non-singular.

2) The matrix is
[1 -2 1]
[5 -1 3]
[4 1 2], which is singular (rank = 2) so the range is not R^3. I don't know how to find a vector not in the range unless I know x_1, x_2, and x_3. Sorry.

However, if x_1, x_2, x_3 are the standard unit vectors [1 0 0], [0 1 0], [0 0 1], then we can say, for example that [1 1 1] is not in the range. If we try to solve the system with augmented matrix
[1 -2 1 1]
[5 -1 3 1]
[4 1 2 1] we get the row echelon form

[1 -2 1 1]
[0 9 -2 -4]
[0 0 0 1]., which of course shows there is no solution.

Answer 2

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