Test 1 Practice

  1. Find if invertible, then find inverse (2x2 matrix)
    det(A)= ad-bc

    A⁻¹= 1/det(A)*[Matrix] (In the matrix swap places of a and d. Change the sign for b an c)
  2. Find if invertible, then find inverse (3x3)
    +- +
    2 4 1
    5 3 2
    3 1 0


    • 1. Find det(A)= 2[Matrix] - 4[Matrix] + 1[Matrix]
    • = Anything nonzero is invertible (Lets say it is 4)

    • 2. Take det(A) of all matrices and make sure the middle X values are positive and the others are negative
    • [Matrix] -[Matrix] [Matrix]
      [-Matrix] [Matrix] -[Matrix]
      [Matrix] [-Matrix] [Matrix]

    • 3. Find Adj(A)
    • Take the top 3 values from the last step and put them on the left most columns (going down now instead of left to right) and fill out.
    • 1 2 3
      4 5 6
      7 8 9
    • --> 
    • 1 4 7
      2 5 8
      3 6 9

    • 4. Find A⁻¹
    • A⁻¹= 1/det(A) [Adj(A) Matrix]

    Solve.
  3. Find all matrices that commute with the given matrix A
    • Solve for:
    • A*B and B*A.

    Set equal to each other

    Solve for a,b,c,d
  4. Find the matrix of a rotation through an angle of 60° in the counterclockwise direction
    • Clockwise: R(-θ)= [cosθ  sinθ]
    •                            [-sinθ cosθ]

    • Counterclockwise: R(θ)= [cosθ  -sinθ]
    •                                      [sinθ   cosθ]

    60°= π/3 = cosθ= (1/2) and sinθ= (-√3/2)

    • So, R(θ)= [cos(π/3)  -sin(π/3)]
    •                [sin(π/3)    cos(π/3)]
    • = [1/2    -√3/2]
    •    [√3/2     1/2]
  5. Clockwise vs Counterclockwise
    • Clockwise: R(-θ)= [cosθ  sinθ]
    •                            [-sinθ cosθ]

    • Counterclockwise: R(θ)= [cosθ  -sinθ]
    •                                      [sinθ   cosθ]

    • Clockwise, the negative is bottom left.
    • Counter CW, the negative is top right.
  6. Let L be the line in R^3 that consists of all scalar multiples of the vector [2 1 2] (going down, not horizontal). Find the orthogonal projection, of the vector [1 1 1] (also going down) about the line L
    Solve for ProjL(v)= (v*a / a*a) * a

    • a= vector youre projecting onto
    • v= any other vector

    • v*a= [2 1 2]*[1 1 1]
    • = 2+1+2 
    • = 5

    • a*a= [2 1 2]*[2 1 2]
    • = 4+1+4
    • 9

    (v*a / a*a) = 5/9 [2 1 2] (going down still)
  7. Find the matrices of the linear transformations from R³ to R³. Assume the transformation is linear:

    The orthogonal projection onto the x-y plane
    • Since the rank is 3, we have x, y, and z. Since the projection is only on the x-y plane, we can say that the z plane is 0.
    • So, v= [x y 0]

    • So, A= [1 0 0]
    •            [0 1 0]
    •            [0 0 0]

    • Therefore, A*v= [1 0 0]   [x]    [x]
    •                         [0 1 0] * [y] = [y]
    •                         [0 0 0]    [0]    [0]

    • So the matrix is: 
    • [1 0 0]
    • [0 1 0]
    • [0 0 0]
  8. Suppose the line L in R^3 contains the unit vector u=[u1,u2,u3]. find the matrix A of the linear transformation T(x)=refL(x). Give the entries of A in terms of the components u1,u2,u3 of u.
    A= 2(u*u)* I2

    •   [u1]                       [u1²    u1u2    u1u3]
    •   [u2] * [u1 u2 u3]= [u1u2   u2²     u2u3]
    •   [u3]                       [u1u3   u2u3   u3²  ]

    • Then multiply all by 2 and subtract by I3
    • [2u1²    2u1u2    2u1u3]   [1 0 0]
    • [2u1u2   2u2²     2u2u3] - [0 1 0]
    • [2u1u3   2u2u3   2u3²  ]   [0 0 1]

    • A= [2u1²-1    2u1u2    2u1u3]
    •      [2u1u2   2u2²-1     2u2u3]
    •      [2u1u3   2u2u3      2u3²-1]
  9. Consider the linear transformation T from R3 to R2 with:
    T[1 0 0]= [7 11], T[0 1 0]= [6 9], and T[0 0 1]= [-13 17]. Find the matrix A of T.

    (All matrices are vertical)
    Tx= Ax

    • A= [7  6 -13]
    •      [11 9 17]
  10. Suppose v1, v1, ... , vm are arbitrary vectors in R^n. Consider the transformation from R^m to R^n given by:
      [x1 ]
    T|x2 | = x1*v1+x2*v2+...+xm*vm
      |... |
      [xm]

    Is the transformation linear? If so, find its matrix A in terms of the vectors v1, v2,..., vm.
    • Tx= Ax
    •                            [x1 ]
    • = [v1 v2 ... vm] * |x2 |
    •                            |... |
    •                            [xm]

    A= [v1 v2 ... vm], Yes it is linear
  11. Find all solutions of the linear system
    \x+2y= a|, where a & b are arbitrary constants .
    |3x+5y= b|
    • [1 2 a] (rref)  .....        [1 0 -5a+2b]
    • [3 5 b] ----->       ---> [0 1   -b+3a]

    • x= -5a+2b
    • y= -b+3a
  12. The matrix [-0.8   -0.6] 
                     [0.6    -0.8] 
    represents a rotation. Find the angle of rotation (in radians).
    • R(Θ)= [-0.8   -0.6]  = cosΘ= -0.8 and sinΘ= 0.6
    •           [0.6    -0.8] 

    • tan-1(sinΘ/cosΘ) + π
    • = tan-1(-0.8/0.6+ π = 2.49
  13. Let L be the line in R³ that consists of all scalar multiples of [2 1 2] (going down). Find the refection vector [1 1 1] (going down) about the L.
    Reflection of v= vₗₗ - v┴

    • vₗₗ= ((v*a) / (a*a))*a
    • and
    • v┴= v-vₗₗ

    • 1. Calculate vₗₗ 
    • ([1 1 1]*[1 2 1] / [2 1 2]*[2 1 2]) * [2 1 2]
    • = (1+2+1) / (4+1+4)
    • =5/9 * [2 1 2] 
    • = [10/9  5/9  10/9]

    • 2. Now, calculate v┴
    • v┴= v-vₗₗ
    • = [1 1 1] - [10/9  5/9  10/9]
    • = [1 - (10/9), 1 - (5/9), 1 - (10/9)]
    • = [-1/9,  4/9,  -1/9]

    • 3. Finally, calculate the reflection
    • v= vₗₗ - v┴
    • = [10/9,  5/9,  10/9] - [-1/9,  4/9,  -1/9]
    • = [11/9,  1/9,  11/9]
  14. Equation to find the orthogonal projection of a vector
    projL(v)= (v*a / a*a) *a
  15. Equation to find the reflection of a vector
    RefL(v)= vₗₗ - v┴

    • where vₗₗ= (v*a / a*a) *a
    • and 
    • where v┴= v-vₗₗ
Author
GoBroncos
ID
365848
Card Set
Test 1 Practice
Description
Updated