## 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 normal 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

- 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.
- 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 .
- 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 .
- 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 .

- 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)

In Example 1, there is a typo in “In fact, we have e’_3=-e’_1”. I believe this should say “e’_3=-e’_2”.

Yes.. thank you. This has been corrected!