Exercise Find the corresponding transformations, and indicate the source and target euclidean spaces.
[ | 1 | 2 | ] |
[ | 1 | ] |
2 |
[ | 2 | 0 | 2 | 1 | ] |
1 | 1 | -1 | 0 | ||
0 | 1 | -2 | 1 |
[ | 1 | 2 | 1 | ] |
2 | 1 | 1 | ||
0 | 3 | 1 | ||
3 | 0 | 1 |
[ | 1 | -1 | 1 | 0 | 2 | ] |
2 | -2 | 0 | 2 | 2 | ||
-1 | 1 | 2 | -3 | 1 | ||
-2 | 2 | 1 | -3 | -1 |
Exercise Which of the following are matrix transformations? For the ones that are, find the corresponding matrices. For the ones that are not, provide the reason.
T_{1}(x_{1}, x_{2}, x_{3}) = (2x_{1} + 3x_{2} - 7x_{3}, 10x_{1} - 4x_{2} - 8x_{3}, - 4x_{1} + x_{2})
T_{2}(x_{1}, x_{2}, x_{3}) = (2x_{1} + 3x_{2} - 7x_{3} + 1, 10x_{1} - 4x_{2} - 8x_{3}, - 4x_{1} + x_{2} - 5)
T_{3}(x_{1}, x_{2}, x_{3}) = 2x_{1} + 3x_{2} - 7x_{3}
T_{4}(x_{1}, x_{2}, x_{3}) = 2x_{1} + 3x_{2} - 7x_{3} + 1
T_{5}(x_{1}, x_{2}, x_{3}) = (2x_{1}, 3x_{2}, - 7x_{3})
T_{6}(x_{1}, x_{2}, x_{3}) = (2x_{2}, 3x_{3}, - 7x_{1})
T_{7}(x_{1}, x_{2}, x_{3}) = (3x_{2}, - 7x_{3}, 2x_{1})
T_{8}(x) = (2x, 3x, - 7x)
T_{9}(x, y, z) = (x - y, y - z, z - x)
T_{10}(x, y) = (x + y, xy)
T_{11}(x, y) = (x + y, 2^{1/2}x + 3^{1/2}y)
T_{12}(u, v) = (u + v, u^{2} + v^{2}, u^{3} + v^{3})
Exercise Convince yourself that the following transformations are linear by drawing the picture and indicating how the parallelogram and scaling by 1/2 are preserved by the transformations. Then find the corresponding matrices.
Reflection in diagonal line: R^{2} → R^{2}
Projection to x_{1}-line: R^{2} → R^{1}
Multiplying by 2: R^{2} → R^{2}
Embedding as diagonal: R^{1} → R^{2}
Reflection in origin: R^{2} → R^{2}
Reflection in origin: R^{3} → R^{3}
Exercise The values of the following linear transformations at the standard basis have been given. Find the value at the other vectors.
T: R^{2} → R^{2}, T(e_{1})
= (5, -3), T(e_{2}) = (-4, 2)
find T(2, 3), T(-3, 1), T(x_{1},
x_{2})
S: R^{2} → R^{3}, S(e_{1})
= (1, 2, 3), S(e_{2}) = (4, 5, 6)
find S(2, 3), S(-3, 1), S(x_{1},
x_{2})
R: R^{3} → R^{2}, R(e_{1})
= (1, 2), R(e_{2}) = (3, 4), R(e3_{})
= (5, 6)
find R(1, 0, 2), R(-2, 1, 3), R(x_{1},
x_{2}_{}, x_{3})
Exercise Let T: R^{2} → R^{3} be a linear transformation. Assume we already know T(1, 2) = (1, 2, 3), T(3, 4) = (4, 5, 6).
Exercise Let T: R^{3} → R^{2} be a linear transformation. Assume we already know T(1, 1, 1) = (1, 2), T(1, 1, 0) = (3, 4), T(1, 0, 0) = (5, 6). Find the general formula (or the matrix) for T.
Exercise Solve the following three systems of linear equations.
x_{1} | + 2x_{2} | = | 1 | x_{1} | + 2x_{2} | = | 0 | |
2x_{1} | + 3x_{2} | = | 0 | 2x_{1} | + 3x_{2} | = | 1 |
Then use your results and the linearity to find solutions of the following systems
x_{1} | + 2x_{2} | = | 2 | x_{1} | + 2x_{2} | = | 13 | |
2x_{1} | + 3x_{2} | = | -3 | 2x_{1} | + 3x_{2} | = | 9 |
Finally use the same technique to find the solution of the system
x_{1} | + 2x_{2} | = | b_{1} |
2x_{1} | + 3x_{2} | = | b_{2} |
Exercise Solve the following three systems of linear equations.
x_{1} | + 2x_{3} | = | 1 | x_{1} | + 2x_{3} | = | 0 | x_{1} | + 2x_{3} | = | 0 | |||||
2x_{1} | - x_{2} | + 3x_{2} | = | 0 | 2x_{1} | - x_{2} | + 3x_{2} | = | 1 | 2x_{1} | - x_{2} | + 3x_{2} | = | 0 | ||
4x_{1} | - x_{2} | + 8x_{2} | = | 0 | 4x_{1} | - x_{2} | + 8x_{2} | = | 0 | 4x_{1} | - x_{2} | + 8x_{2} | = | 1 |
Then use your results and the linearity to find a solution of the system
x_{1} | + 2x_{3} | = | b_{1} | |
2x_{1} | - x_{2} | + 3x_{2} | = | b_{2} |
4x_{1} | - x_{2} | + 8x_{2} | = | b_{3} |