Linear Vector Spaces: Examples and Problems
Examples Problems
Example 1
Which of the following sets are orthonormal basis sets in the Euclidean vector space ?
where
,
, and
.
where
,
, and
.
Solution
For the set we have:
Therefore, ,
, and
are unit vectors. In addition, we have:
Therefore, the three vectors ,
, and
are orthogonal to each other. Therefore,
is an orthonormal basis set.
For the set we have:
Therefore, and
are not orthogonal to each other. Therefore,
is not an orthonormal basis set. In fact, we have
which means that
is linearly dependent, therefore, it is not even a basis set for
. Notice that the symbol “.” is used in Mathematica for the dot product as shown below.
View Mathematica Code:
e1 = {1, 0, 0}; e2 = {0, 1/Sqrt[2], 1/Sqrt[2]}; e3 = {0, 1/Sqrt[2], -1/Sqrt[2]}; e1.e1 e1.e2 e1.e3 e2.e2 e2.e3 e3.e3 ep1 = {1, 0, 0}; ep2 = {0, 1/Sqrt[2], 1/Sqrt[2]}; ep3 = {0, -1/Sqrt[2], -1/Sqrt[2]}; ep1.ep1 ep1.ep2 ep1.ep3 ep2.ep2 ep2.ep3 ep3.ep3
Example 2
Find the angle between the two vectors , and
. Also use the cross product operation to find the vector
.
Solution
To find the geometric angle between the two vectors
and
, we will use the dot product operation:
We have ,
and
. Therefore,
and
are orthogonal and
.
The cross product between and
gives the vector
as follows:
x = {1, 1, 1}; y = {-1, 2, -1}; z = Cross[x, y] Norm[x] Norm[y] thetaxy = ArcCos[x.y/Norm[x]/Norm[y]] (*You can also find the angle between the two vectors using the command VectorAngle*) VectorAngle[x,y]
Problems
- Show that the following sets are subspaces of
. What is the graphical representation of each subspace? Find two different basis sets for each subspace.
.
.
- Show that the following sets are subspaces of
. What is the graphical representation of each subspace? Find two different basis sets for each subspace.
.
.
- Find three vectors that are orthogonal to
.
- Find three vectors that are orthogonal to
.
- Choose a value for
such that
, and
are linearly independent.
- Choose two different sets of values for
and
such that
, and
are linearly dependent.
- Verify that
, and
are linearly independent and then find the unique expansion of
in the basis set
.
- Verify that
, and
are linearly independent and then find the unique expansion of
in the basis set
.
- Find two different orthonormal basis sets and two different non-orthonormal basis sets for
.
- Find two different orthonormal basis sets and two different non-orthonormal basis sets for
.
- Find two different orthonormal basis sets and two different non-orthonormal basis sets for
. The basis sets should include vector
. Hint: An orthonormal basis set is a basis set whose vectors satisfy two conditions. The first condition is that the vectors in the basis set are orthogonal to each other and the second condition is that each vector has a unit norm.
- Show that the following vectors are linearly dependent
,
, and
.
- Show that the following vectors are linearly dependent
,
, and
.
- Verify that the following vectors are linearly independent
,
, and
. Then, find the unique expansion of
in the basis set
.
- Verify that the following vectors are linearly independent
,
, and
. Then, find the unique expansion of
in the basis set
.
- Use the cross product to find a vector orthogonal to both
and
. Also, find the area of the parallelogram formed by the two vectors
and
.
- Use the cross product to find a vector orthogonal to both
and
. Also, find the area of the parallelogram formed by the two vectors
and
.
- For the shown cuboid,
,
,
, and
.
- Use the cross product operation to find two unit vectors orthogonal to the plane
. What is the relationship between those two vectors?
- Use the cross product to find the area of the parallelogram
.
- Find the angle between the vectors representing
and
.
Notice that the line geometric object starting at point
and ending at point
can be represented by a vector
.
- Use the cross product operation to find two unit vectors orthogonal to the plane
- Determine if the following vectors
and
are linearly independent where
and
while
is a set of linearly independent vectors in
.
- Which of the following functions
defined below satisfy the properties of a norm function (explain your answer).
:
.
.
.
.
- Let
. Assume that
. Show that
is the zero vector. (Hint: Show that the components of
in an orthonormal basis set are all equal to zero)