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Transforming Vectors with Matrices

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Transforming Vectors with Matrices Answer Key

Instructions: Unless otherwise stated, vectors are represented as column matrices.

1. 
Let [math]vec{v}[/math] be a vector with 2 or more components and represented as a column matrix, and let [math]A[/math] be a transformation matrix. When the transformation matrix is applied to [math]vec{v}[/math] it results in a new vector, [math]vec{w}[/math]. Which of the following correctly represents this?
  1. [math]vec{w} = vec{v}A[/math]
  2. [math]vec{w} = A vec{v}[/math]
  3. [math]vec{w} = vec{v} A vec{v} [/math]
  4. Both options (a) and (b) are correct.
2. 
How would the vector [math]<< 1,4,7 >>[/math] be represented as a matrix?
  1. [math][[1,4,7],[0,0,0],[0,0,0]][/math]
  2. [math][[1],[4],[7]][/math]
  3. [math][[7],[4],[1]][/math]
  4. [math][ [1,0,0],[0,4,0],[0,0,7]][/math]
3. 
How would the vector [math]<< 4,10,-5 >>[/math] would be represented as a matrix?
  1. [math][[4,10,-5],[0,0,0],[0,0,0]][/math]
  2. [math][ [4,0,0],[0,10,0],[0,0,-5]][/math]
  3. [math][[-5],[10],[4]][/math]
  4. [math][[4],[10],[-5]][/math]
4. 
Find the product of [math][[3,-1],[4,-2]][/math] and [math]<<4,-3>>[/math].
  1. These cannot be multiplied together.
  2. [math][[9],[10]][/math]
  3. [math][[0,2]][/math]
  4. [math][[15],[22]] [/math]
5. 
For the vector [math]<<3,-1,4>> [/math], find the resulting vector if the transformation matrix [math][[3,2,1],[3,2,1],[3,2,1]][/math] is applied to it.
  1. [math][[11],[11],[11]][/math]
  2. [math][[18],[12],[6]][/math]
  3. [math][[13],[13],[13]][/math]
  4. This vector and matrix cannot be multiplied together.
6. 
What does the matrix [math][[2,6,1],[4,1,8]][/math] multiplied by the vector [math]<< 4,10,-5 >>[/math] equal?
  1. [math][[8,60,-5],[16,10,-40],[0,0,0]][/math]
  2. [math][ [16,10,-40],[8,60,-5]][/math]
  3. [math][[24],[70],[-45]][/math]
  4. [math][[63],[-14]][/math]
7. 
The matrix [math][[1,2,3],[4,5,6],[7,8,9]][/math] multiplied by the vector [math]<< 1,2,3 >>[/math] equals which of the following?
  1. [math][[1,4,9],[4,10,18],[7,16,27]][/math]
  2. [math][ [14],[32],[50]][/math]
  3. [math][[12],[30],[54]][/math]
  4. [math][[1,2,3],[8,10,12],[21,24,27]][/math]
8. 
Which matrix would transform the vector [math]<< 4,2,7 >>[/math] to the vector [math]<<20,23,51 >>[/math]?
  1. [math][[0,3,2],[4,0,1],[3,2,5]][/math]
  2. [math][[3,2,5],[4,0,1],[0,3,2]][/math]
  3. [math][[0,4,3],[3,0,2],[2,1,5]][/math]
  4. [math][[5,1,2],[2,0,3],[3,4,0]][/math]
9. 
For the transformation matrix [math]A = [[1,0],[0,-1]][/math], how is the vector [math]vec{v} = <<3, 4>>[/math] affected if it is multiplied with [math]A ?[/math]
  1. It has been rotated 90° clockwise.
  2. It has reversed direction.
  3. It has been reflected over a horizontal line.
  4. The resulting transformation is a combination of reflections and rotations.
10. 
For the matrix [math]A = [[1,0,0],[0,1,0],[0,0,0]][/math], what is the best description of how this transforms a vector with 3 components if they are multiplied together?
  1. The vector stays the same.
  2. The vector now only has 2 components.
  3. The vector's third component is changed to zero.
  4. At least one component of the vector is reduced to zero.

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