## 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:

View Mathematica Code:

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

1. Show that the following sets are subspaces of . What is the graphical representation of each subspace? Find two different basis sets for each subspace.
• .
• .
2. Show that the following sets are subspaces of . What is the graphical representation of each subspace? Find two different basis sets for each subspace.
• .
• .
3. Find three vectors that are orthogonal to .
4. Find three vectors that are orthogonal to .
5. Choose a value for such that , and are linearly independent.
6. Verify that , and are linearly independent and then find the unique expansion of in the basis set .
7. Find two different orthonormal basis sets and two different non-orthonormal basis sets for .
8. Find two different orthonormal basis sets and two different non-orthonormal basis sets for .
9. Show that the following vectors are linearly dependent , , and .
10. Verify that the following vectors are linearly independent , , and . Then, find the unique expansion of in the basis set .
11. 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 .
12. 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 .

13. Which of the following functions defined below satisfy the properties of a norm function (explain your answer). :
• .
• .
• .
• .
14. 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)