Linear Algebra I: Homework 4

Due: Friday, September 15, 2017
  1. Let be the vector in which points from to .
    1. Express as a column vector.

    2. Find the magnitude of .

    3. Find the angle from to the vector,

      Because , we get that

  2. The matrix

    has a nice graphical explanation. If is a 2-vector in , the vector (that is, the product of the matrix multiplication) has the same length as , but has been rotated by degrees counterclockwise ( can be any angle).

    Let be the 2-vector

    Without actually computing the vector , compute the following:

    1. The dot product,

      The vector is the same as but rotated by radians. So, it is orthogonal to , meaning that the dot product is .

    2. The magnitude,

      The vector is the same as but rotated by radians. So, it is the same length as , and its magnitude .

  3. Let be a fixed vector in . For each part, describe in words the set of all vectors that satisfy the stated condition. Hint: Think about nice shapes. An answer which just re-writes the math in English will not receive full credit.

    1. .

      If the vector is drawn starting at the origin and ending at a point , then the set of all is the set of all vectors which, when drawn starting at he origin, end in the unit circle around .

    2. .

      If the vector is drawn starting at the origin and ending at a point , then the set of all is the set of all vectors which, when drawn starting at he origin, end in or outside of the unit circle around .

  4. Explain why the line of 3-vectors,

    is not a vector space.

    You only have to show that fails one of the vector space axioms. The easiest in this case is that it fails to have the zero vector :

    Vectors in look like

    and always have a 1 in the third component. But the zero vector is

    which has a zero (not a 1) in the third component, and so can’t be in .

  5. Does there exist a linear transformation such that,

    and,

    Justify your answer.

    No. If were linear then

    But this is different from what the problem says it should be:

    So can’t be linear.