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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 $vec{v}$ be a vector with 2 or more components and represented as a column matrix, and let $A$ be a transformation matrix. When the transformation matrix is applied to $vec{v}$ it results in a new vector, $vec{w}$. Which of the following correctly represents this?
1. $vec{w} = vec{v}A$
2. $vec{w} = A vec{v}$
3. $vec{w} = vec{v} A vec{v}$
4. Both options (a) and (b) are correct.
2.
How would the vector $<< 1,4,7 >>$ be represented as a matrix?
1. $[[1,4,7],[0,0,0],[0,0,0]]$
2. $[[1],[4],[7]]$
3. $[[7],[4],[1]]$
4. $[ [1,0,0],[0,4,0],[0,0,7]]$
3.
How would the vector $<< 4,10,-5 >>$ would be represented as a matrix?
1. $[[4,10,-5],[0,0,0],[0,0,0]]$
2. $[ [4,0,0],[0,10,0],[0,0,-5]]$
3. $[[-5],[10],[4]]$
4. $[[4],[10],[-5]]$
4.
Find the product of $[[3,-1],[4,-2]]$ and $<<4,-3>>$.
1. These cannot be multiplied together.
2. $[[9],[10]]$
3. $[[0,2]]$
4. $[[15],[22]]$
5.
For the vector $<<3,-1,4>>$, find the resulting vector if the transformation matrix $[[3,2,1],[3,2,1],[3,2,1]]$ is applied to it.
1. $[[11],[11],[11]]$
2. $[[18],[12],[6]]$
3. $[[13],[13],[13]]$
4. This vector and matrix cannot be multiplied together.
6.
What does the matrix $[[2,6,1],[4,1,8]]$ multiplied by the vector $<< 4,10,-5 >>$ equal?
1. $[[8,60,-5],[16,10,-40],[0,0,0]]$
2. $[ [16,10,-40],[8,60,-5]]$
3. $[[24],[70],[-45]]$
4. $[[63],[-14]]$
7.
The matrix $[[1,2,3],[4,5,6],[7,8,9]]$ multiplied by the vector $<< 1,2,3 >>$ equals which of the following?
1. $[[1,4,9],[4,10,18],[7,16,27]]$
2. $[ [14],[32],[50]]$
3. $[[12],[30],[54]]$
4. $[[1,2,3],[8,10,12],[21,24,27]]$
8.
Which matrix would transform the vector $<< 4,2,7 >>$ to the vector $<<20,23,51 >>$?
1. $[[0,3,2],[4,0,1],[3,2,5]]$
2. $[[3,2,5],[4,0,1],[0,3,2]]$
3. $[[0,4,3],[3,0,2],[2,1,5]]$
4. $[[5,1,2],[2,0,3],[3,4,0]]$
9.
For the transformation matrix $A = [[1,0],[0,-1]]$, how is the vector $vec{v} = <<3, 4>>$ affected if it is multiplied with $A ?$
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 $A = [[1,0,0],[0,1,0],[0,0,0]]$, 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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