Composition and Matrix Multiplication

Exercise Compute the composition TS.

T(y) = 1 - y2: RR
S(x) = sin x: RR

T(x, y) = x2 + y2: R2R
S(t) = (cos t, sin t): RR2

T(x, y) = (x + y, x - y): R2R2
S(u, v) = (1/2(u + v), 1/2(u - v)): R2R2

T(x1, x2, x3, x4, x5) = (x1 - 2x2 + 3x3 - 4x4 + 5x5): R5R
S(t) = (8t, 7t, 6t, 5t, 4t): RR5

T(t) = (8t, 7t, 6t, 5t, 4t): RR5
S(x1, x2, x3, x4, x5) = (x1 - 2x2 + 3x3 - 4x4 + 5x5): R5R

T(x1, x2, x3) = (x1 - 2x2 + 3x3, - 2x1 + 3x2 - 4x3): R3R2
S(y1, y2) = (- 2y1 - y2, y2, y1 + y2): R2R3

T(y1, y2) = (- 2y1 - y2, y2, y1 + y2): R2R3
S(x1, x2, x3) = (x1 - 2x2 + 3x3, - 2x1 + 3x2 - 4x3): R3R2

Exercise Draw the picture and identify the composition TS.

T = Reflection in x-axis: R2R2
S = Reflection in y-axis: R2R2

T = Rotation by 90 degrees: R2R2
S = Reflection in origin: R2R2

T = Reflection in origin: R2R2
S = Rotation by 90 degrees: R2R2

T = Reflection in x-axis: R2R2
S = Embedding as diagonal: RR2

T = Embedding as diagonal: RR2
S = Projection onto x-axis: R2R

T = Projection onto x-axis: R2R
S = Embedding as diagonal: RR2

Exercise Compute matrix products.

 [ 1 1 ][ 1/2 1/2 ] 1 -1 1/2 -1/2
 [ ] [ 8 ] 7 1 -2 3 -4 5 6 5 4
 [ 8 ] [ ] 7 6 1 -2 3 -4 5 5 4
 [ 1 -2 3 ] -2 3 -4
 [ -2 -1 ] 0 1 1 1
 [ -2 -1 ] 0 1 1 1
 [ 1 -2 3 ] -2 3 -4
 [ -2 -1 ][ 1 -2 3 ] 0 1 -2 3 -4
 [ 1 2 3 4 ] 5 6 7 8 9 10 11 12
 [ 1 1 ] -1 -1 1 -1 -1 1
 [ 1 -1 1 ][ 1 2 3 4 ] -1 2 -1 5 6 7 8 2 0 -1 9 10 11 12

Exercise For each of the following matrix A, find all the matrices X satisfying AX = XA.

 [ 1 2 ]
 [ 1 1 ] 1 -1
 [ 1 0 ] 0 2
 [ 0 0 ] 0 0
 [ 2 0 ] 0 2
 [ 2 0 0 ] 0 2 0

Exercise For each of the following matrix A, find all the two column matrices X satisfying AX = O (the zero matrix).

 [ 1 2 ]
 [ 1 ] 2
 [ 1 1 ] 1 -1
 [ 2 4 ] -1 -2
 [ 0 0 ] 0 0
 [ 1 -2 3 ] -2 3 -4
 [ 1 -2 ] -2 3 3 -4

Exercise For each of the following matrix A, find all the matrices X satisfying AX = I (the identity matrix).

 [ 1 2 ]
 [ 1 ] 2
 [ 1 1 ] 1 -1
 [ 2 4 ] -1 -2
 [ 0 0 ] 0 0
 [ 1 -2 3 ] -2 3 -4
 [ 1 -2 ] -2 3 3 -4

Exercise Let

 A = [ 2 0 2 1 ] 1 1 -1 0 0 1 -2 1

For each of the following matrix B, find a matrix X such that AX = B.

 [ 1 ] -1 -2
 [ 1 1 ] -1 0 -2 1
 [ 1 0 0 ] 0 1 0 0 0 1

Exercise Let

 A = [ 1 -1 1 0 2 ],     B = [ 1 3 1 ] 2 -2 0 2 2 0 2 0 -1 1 2 -3 1 0 3 2 -2 2 1 -3 -1 0 0 1

Can you find a matrix X such that AX = B? In general, what is the criterion for the existence of X such that AX = B? What about the uniqueness of X?

Exercise Describe the first two rows and the last three columns of AB in terms of row and columns of A and B.

Exercise The trace trA of a square matrix is the sum of its diagonal entries. For example,

 tr[ 1 2 ] = 1 + 4 = 5 3 4
 tr[ 0 3 -4 ] = 0 + 5 - 3 = 2 2 5 7 0 4 -3

Prove the trace has the following properties

tr(A + B) = trA + trB, tr cA = c trA, trAT = trA, trAB = trBA.

Exercise What does the multiplication of two upper triangular matrices look like? What about lower triangular matrices? diagonal matrices?

Exercise What are the matrices A satisfying AX = XA for any X?

Exercise

1. Prove that transformations T and S are onto ⇒ The composition TS is onto.
2. What about the converse? In other words, if the composition TS is onto, are the transformations T and S onto?
3. Rephrase your conclusions in the first and second part in terms of matrix products.