Express as a column vector.
Find the magnitude of .
Find the angle from to the vector,
Because , we get that
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:
The dot product,
The vector is the same as but rotated by radians. So, it is orthogonal to , meaning that the dot product is .
The magnitude,
The vector is the same as but rotated by radians. So, it is the same length as , and its magnitude .
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.
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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 .
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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 .
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 .
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.