Solving A x = y by the numbers

Here's the picture of xA x for the 2 × 2 matrix A = (v v ) (s ) (u ) = ( 3 4 ) (3 0) ( 5 12 ) ... --- -- 5 5 13 13.  

Your job is to solve A x = y = (2) 3.

Step 1:  Find the v_i coordinates for y .  
From the plot it appears that y≈ -1v_1 + 3v_2.
Since {v_1, v_2} = {(3 ), (4)} - - 5 5 4 3 -- - 5 5 is an orthonormal basis, you can compute the exact coordinates with dot products.
    y = (v_1 · y) v_1 + (v_2 · y) v_2 = (6/5 - 12/5) v_1 + (8/5 + 9/5) v_2 = -6/5 v_1 + 17/5v_2.

Step 2:  Divide the -6/5 and 17/5 by the singular values 3 and 2 to get  -3/5, 17/10.

Step 3:  Use these numbers as u_i coordinates for x to get x = -3/5u_1 + 17/10u_2.

Multiply out to get a numerical answer:
    x = -3/5u_1 + 17/10u_2 = x = -3/5 (5 ) + 17/10 (-12) = ( 9 ) ... -- -- 13 13 10.

Final answer:  The solution to A x = (2) 3 is x = ( 9 ) -- 5 1 -- 10.

Here's the picture of xA x for the 2 × 2 matrix A = (v v ) (s ) (u ) = ( 7 24 ) (4 0) ( 3 4 ) ... -- - 25 25 5 5.  

Your job is to solve A x = y = (4) 2.

Step 1:  Find the v_i coordinates for y .  
From the plot it appears that y≈ -1v_1 + 4.5v_2.
Since {v_1, v_2} = {(7 ), (24)} -- -- 25 25 -24 7 --- -- 25 25 is an orthonormal basis, you can compute the exact coordinates by using dot products.

(Q24)  y = ___ v_1 + ___ v_2

Step 2:  Divide the v_i coordinates by the singular values.

Step 3:  Use these numbers as u_i coordinates for x to get

(Q25)  x = ___ u_1 + ___ u_2

Multiply out to get a numerical answer.

(Q26)  The solution to A x = (4) 2 is x = (  )     .

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Copyright © 2007 Todd Will
Created by Mathematica  (April 15, 2007)