### Linear Transformation

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.

T1(x1, x2, x3) = (2x1 + 3x2 - 7x3, 10x1 - 4x2 - 8x3, - 4x1 + x2)

T2(x1, x2, x3) = (2x1 + 3x2 - 7x3 + 1, 10x1 - 4x2 - 8x3, - 4x1 + x2 - 5)

T3(x1, x2, x3) = 2x1 + 3x2 - 7x3

T4(x1, x2, x3) = 2x1 + 3x2 - 7x3 + 1

T5(x1, x2, x3) = (2x1, 3x2, - 7x3)

T6(x1, x2, x3) = (2x2, 3x3, - 7x1)

T7(x1, x2, x3) = (3x2, - 7x3, 2x1)

T8(x) = (2x, 3x, - 7x)

T9(x, y, z) = (x - y, y - z, z - x)

T10(x, y) = (x + y, xy)

T11(x, y) = (x + y, 21/2x + 31/2y)

T12(u, v) = (u + v, u2 + v2, u3 + v3)

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: R2R2

Projection to x1-line: R2R1

Multiplying by 2: R2R2

Embedding as diagonal: R1R2

Reflection in origin: R2R2

Reflection in origin: R3R3

Exercise The values of the following linear transformations at the standard basis have been given. Find the value at the other vectors.

T: R2R2, T(e1) = (5, -3), T(e2) = (-4, 2)
find T(2, 3), T(-3, 1), T(x1, x2)

S: R2R3, S(e1) = (1, 2, 3), S(e2) = (4, 5, 6)
find S(2, 3), S(-3, 1), S(x1, x2)

R: R3R2, R(e1) = (1, 2), R(e2) = (3, 4), R(e3) = (5, 6)
find R(1, 0, 2), R(-2, 1, 3), R(x1, x2, x3)

Exercise Let T: R2R3 be a linear transformation. Assume we already know T(1, 2) = (1, 2, 3), T(3, 4) = (4, 5, 6).

• Use this exercise and the linearity of T to compute T(4, 6), T(1, 1), T(1, 0), T(0, 1).
• Use T(1, 0), T(0, 1) to compute T(x1, x2) in general.
• Use 3(1, 1, 1) = (4, 5, 6) - (1, 2, 3) to find x1, x2 such that T(x1, x2) = (1, 1, 1)

Exercise Let T: R3R2 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.

 x1 + 2x2 = 1 x1 + 2x2 = 0 2x1 + 3x2 = 0 2x1 + 3x2 = 1

Then use your results and the linearity to find solutions of the following systems

 x1 + 2x2 = 2 x1 + 2x2 = 13 2x1 + 3x2 = -3 2x1 + 3x2 = 9

Finally use the same technique to find the solution of the system

 x1 + 2x2 = b1 2x1 + 3x2 = b2