Samer Adeeb

Linear Vector Spaces: Examples and Problems

Examples Problems

Example 1

Which of the following sets are orthonormal basis sets in the Euclidean vector space \mathbb{R}^3?

  • B_1=\{e_1,e_2,e_3\} where e_1=\{1,0,0\}, e_2=\left\{0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right\}, and e_3=\left\{0,\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}}\right\}.
  • B_2=\{e'_1,e_2,e_3\} where e'_1=\{1,0,0\}, e'_2=\left\{0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right\}, and e'_3=\left\{0,\frac{-1}{\sqrt{2}},\frac{-1}{\sqrt{2}}\right\}.
Solution

For the set B_1 we have:

    \[ e_1\cdot e_1=1\qquad e_2\cdot e_2=\sqrt{\left(\frac{1}{\sqrt{2}}\right)^2+\left(\frac{1}{\sqrt{2}}\right)^2}=1 \qquad e_3\cdot e_3=\sqrt{\left(\frac{1}{\sqrt{2}}\right)^2+\left(\frac{1}{\sqrt{2}}\right)^2}=1 \]

Therefore, e_1, e_2, and e_3 are normal vectors. In addition, we have:

    \[ e_1\cdot e_2=0\qquad e_1\cdot e_3=0 \qquad e_2\cdot e_3=\frac{1}{\sqrt{2}}\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\frac{-1}{\sqrt{2}}=0 \]

Therefore, the three vectors e_1, e_2, and e_3 are orthogonal to each other. Therefore, B_1 is an orthonormal basis set.
For the set B_2 we have:

    \[ e'_2\cdot e'_3=\frac{1}{\sqrt{2}}\frac{-1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\frac{-1}{\sqrt{2}}=-1 \]

Therefore, e'_2 and e'_3 are not orthogonal to each other. Therefore, B_2 is not an orthonormal basis set. In fact, we have e'_3=-e'_2 which means that B_2 is linearly dependent, therefore, it is not even a basis set for \mathbb{R}^3. 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 x=\{1,1,1\}, and y=\{-1,2,-1\}. Also use the cross product operation to find the vector z=x\times y.

Solution

To find the geometric angle \theta_{xy} between the two vectors x and y, we will use the dot product operation:

    \[ \theta_{xy}=\arccos{\left(\frac{x\cdot y}{\|x\| \|y\|}\right)} \]

We have x\cdot y=-1+2-1=0, \|x\|=\sqrt{3} and \|y\|=\sqrt{6}. Therefore, x and y are orthogonal and \theta_{xy}=\frac{\pi}{2}.
The cross product between x and y gives the vector z as follows:

    \[ z=\{-1-2,-1+1,2+1\}=\{-3,0,3\} \]

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 \mathbb{R}^2. What is the graphical representation of each subspace? Find two different basis sets for each subspace.
    • Y=\left\{\alpha x\bigg| x=\left(\begin{array}{c}1\\0\end{array}\right), \alpha\in\mathbb{R}\right\}=\mbox{span}\{x\}.
    • Z=\left\{\alpha z\bigg| z=\left(\begin{array}{c}1\\1\end{array}\right), \alpha\in\mathbb{R}\right\}=\mbox{span}\{z\}.
  2. Show that the following sets are subspaces of \mathbb{R}^3. What is the graphical representation of each subspace? Find two different basis sets for each subspace.
    • Y=\left\{\alpha x\bigg| x=\left(\begin{array}{c}1\\0\\0\end{array}\right), \alpha\in\mathbb{R}\right\}=\mbox{span}\{x\}.
    • Z=\left\{\alpha x+\beta y\bigg| x=\left(\begin{array}{c}1\\0\\0\end{array}\right), y=\left(\begin{array}{c}0\\1\\0\end{array}\right),\alpha,\beta\in\mathbb{R}\right\}=\mbox{span}\{x,y\}.
  3. Find three vectors that are orthogonal to x=\{1,1\}.
  4. Find three vectors that are orthogonal to x=\{1,1,1\}.
  5. Choose a value for y_2 such that x=\{1,1\}, and y=\{1,y_2\} are linearly independent.
  6. Verify that x=\{1,0\}, and y=\{1,2\} are linearly independent and then find the unique expansion of v=\{5,3\} in the basis set B=\{x,y\}.
  7. Find two different orthonormal basis sets and two different non-orthonormal basis sets for \mathbb{R}^2.
  8. Find two different orthonormal basis sets and two different non-orthonormal basis sets for \mathbb{R}^3.
  9. Show that the following vectors are linearly dependent x=\{1,1,1\}, y=\{1,2,1\}, and z=\{0,-1,0\}.
  10. Verify that the following vectors are linearly independent x=\{1,0,1\}, y=\{1,2,1\}, and z=\{0,-1,1\}. Then, find the unique expansion of v=\{5,3,2\} in the basis set B=\{x,y,z\}.
  11. Use the cross product to find a vector orthogonal to both x=\{1,1,1\} and y=\{1,2,0\}. Also, find the area of the parallelogram formed by the two vectors x and y.
  12. For the shown cuboid, A=\{0,1,0\}, B=\{0,1,1\}, C=\{2,0,1\}, and D=\{2,0,0\}.

    la

    • Use the cross product operation to find two unit vectors orthogonal to the plane ABCD. What is the relationship between those two vectors?
    • Use the cross product to find the area of the parallelogram ABCD.
    • Find the angle between the vectors representing AC and BC.

    Notice that the line geometric object starting at point A and ending at point B can be represented by a vector AB=B-A.

  13. Which of the following functions f_1, f_2,f_3,f_4:\mathbb{R}^n\rightarrow [0,\infty) defined below satisfy the properties of a norm function (explain your answer). \forall x\in\mathbb{R}^n:
    • f_1(x)=\max_{i\leq n}|x_i|.
    • f_2(x)=|x_2|.
    • f_3(x)=\sum_{i=1}^n|x_i|.
    • f_4(x)=\sum_{i=1}^n x_i.
  14. Let u\in\mathbb{R}^n. Assume that \forall v\in\mathbb{R}^n:u\cdot v=0. Show that u is the zero vector. (Hint: Show that the components of u in an orthonormal basis set are all equal to zero)

Page Comments

  1. Vincent Huang says:

    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”.

    1. Samer Adeeb says:

      Yes.. thank you. This has been corrected!

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