Download Linear Algebra 2270 Homework 9 Problems:

Linear Algebra 2270
Homework 9
preparation for the quiz on 07/22/2015
Problems:
1. Consider the following:
⎤
⎡
⎡ ⎤
⎛⎢⎢x1 ⎥⎥⎞ ⎢⎢x1 + x2 + x3 ⎥⎥
TA ∶ R → R , TA ⎜⎢x2 ⎥⎟ = ⎢ x1 + x3 ⎥
⎝⎢⎢x3 ⎥⎥⎠ ⎢⎢ 2x1 + x3 ⎥⎥
⎦
⎣
⎣ ⎦
3
3
matrix B ∈ R
3×3
⎡1 −1 0⎤
⎥
⎢
⎥
⎢
, B = ⎢1 0 0⎥
⎥
⎢
⎢0 0 1⎥
⎦
⎣
⎡ −x2 ⎤
⎥
⎢
x
⎥
⎢
1
TC ∶ R2 → R3 , TC ([ ]) = ⎢x1 + 2x2 ⎥
⎥
⎢
x2
⎢ x1 − x2 ⎥
⎦
⎣
(a) Find matrices of TA , TC in the canonical bases A = M(TA ), C = M(TC ).
(b) Find a formula (similar to formulas for TA , TC above) for the linear operator TB such that
M(TB ) = B.
(c)
i. Using the definition of composition of operators, find formulas for TA ○ TB , (TA ○ TB ) ○ TC ,
TB ○ TC , TA ○ (TB ○ TC ).
ii. Using the formula for TA ○ TB , find matrix M(TA ○ TB ) and using the matrix-matrix
multiplication calculate A ⋅ B. Compare the results.
iii. Using the formula for (TA ○ TB ) ○ TC , find matrix M((TA ○ TB ) ○ TC ) and using the
matrix-matrix multiplication calculate (A ⋅ B) ⋅ C. Compare the results.
iv. By performing matrix-matrix multiplication find (B ⋅ C) and then A ⋅ (B ⋅ C). Compare it
with 1(c)iii.
i. Using the algorithm presented in the lecture for finding the inverse of a matrix, find A−1 .
⎡1⎤
⎢ ⎥
⎢ ⎥
Use A−1 to solve a linear system Ax = ⎢0⎥
⎢ ⎥
⎢2⎥
⎣ ⎦
⎡x1 ⎤
⎡1⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
ii. Find x = ⎢x2 ⎥ such that TA (x) = ⎢0⎥
⎢ ⎥
⎢ ⎥
⎢x3 ⎥
⎢2⎥
⎣ ⎦
⎣ ⎦
(e) i. Using the algorithm for finding the inverse of a matrix presented in the lecture, find A−1 ,
B −1 .
ii. Using the same algorithm, find (A ⋅ B)−1 . (Start with a matrix [A ⋅ B∣I] and by doing
the row reduction arrive at [I∣(A ⋅ B)−1 ]). Then using the matrix-matrix multiplication
calculate B −1 ⋅ A−1 . Compare the results
iii. Using the algorithm for finding the inverse, find (A−1 )−1 (Start with a matrix [A−1 ∣I] and
by doing the row reduction arrive at [I∣(A−1 )−1 ]). Compare (A−1 )−1 it with A.
(d)
i. Use the inverse A−1 to write a formula for TA−1 .
ii. Using the definition of composition of operators, find formulas for TA ○ TA−1 , TA−1 ○ TA and
use those to find matrices M(TA ○ TA−1 ), M(TA−1 ○ TA ).
iii. Using the definition of matrix-matrix multiplication, calculate A−1 ⋅A and A⋅A−1 . Compare
the results.
0 2
(g) Consider a matrix D = [
]. Find D−1 by performing three row operations: r1 ↔ r2 ,
1 2
r2 = 21 r2 , r1 + = (−2)r2 . Find the matrices E1 , E2 , E3 of those elementary row operations.
Using matrix-matrix multiplication calculate E3 ⋅ E2 ⋅ E1 and compare it with D−1 .
(f)
1