A is 2 × 2 matrix s.t. ∀ x ∈ R 2 , A x is vector obtained from projecting x to 2 x 1 − x 2 = 0, give A satisfying the condition as stated aboveWhat I've done so far: A = [ a b c d ] , x = [ y 1 y 2 ] which means we can write the following: [ a b c d ] [ y 1 y 2 ] = [ a y 1 + b y 2 c y 1 + d y 2 ] projection= 2 y 1 − y 2 y 1 2 + y 2 2 &lt; y 1 , y 2 &gt;However, at this point, I feel like I'm understanding the direction in which I should go incorrectly.

Question

A is   2  ×  2 matrix s.t.   ∀  x  ∈      R    2    ,  A  x is vector obtained from projecting x to   2      x    1    −      x    2    =  0, give A satisfying the condition as stated aboveWhat I&#039;ve done so far:  A  =      [                            a                          b                                      c                          d                      ]  ,   x  =      [                                        y            1                                                            y            2                                ]   which means we can write the following:       [                            a                          b                                      c                          d                      ]        [                                        y            1                                                            y            2                                ]    =      [                            a                      y            1                    +          b                      y            2                                                c                      y            1                    +          d                      y            2                                ]  projection=            2              y        1            −              y        2                            y        1        2            +              y        2        2              &amp;lt;      y    1    ,      y    2    &amp;gt;However, at this point, I feel like I&#039;m understanding the direction in which I should go incorrectly.

Szulikto · Accepted Answer

If u is a unit vector, then   f  (  x  )  :=  (      u    ⊤    x  )  u  =  (  u      u    ⊤    )  x is the projection onto the span of u (a one-dimensional subspace). So all you need to do is find a unit vector that lies on the line   {  (      x    1    ,      x    2    )  :  2      x    1    −      x    2    =  0  }

A is 2 xx 2 matrix s.t. AA x in R^2 ,Ax is vector obtained from projecting x to 2x_1−x_2=0, give A satisfying the condition

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