Comment due Sunday, November 25
Test #3 will be given in class Monday, December 2. The format will be similar to the format of Test #1 and Test #2.
Recall from Test #1 and Test #2 that Question #1 asked you a series of conceptual true/false questions where you had to justify your answer. Question #2 asked you for a series of examples of mathematical objects (mostly matrices) satisfying certain conditions.
To prepare for Test #3, for this week’s OpenLab assignment, you will comment on this post with two questions that you come up with yourself, as well as their answers.
- Your first question should be conceptual and phrased as a statement which is either always true or always false. Your answer should indicate whether the statement is true or false together with a sentence explaining the answer.
- Your second question should be asking for an example of a mathematical object satisfying certain conditions. Your answer should provide this example together with together with a sentence explaining the example and why it satisfies the conditions.
You can use the Test #1 and Test #2 questions for inspiration (the different versions of the tests had similar questions, so check out your classmates’ solutions here and here for the other versions).
Try to focus on the material covered in class since Test #2. You can see the list of topics on the schedule.
Question 1: Is it true or false that the dimensions of a set depends on how many vectors are currently within it?
Answer: Yes it’s true. This is because another word to find the dimensions is “rank”. To find the rank of a set you have to see what the basis is.
Question 2: Given the vectors u=<1,4>, v=<2,3>, and w=<3,2>. D
Question 2: Given the vectors u=<1,4>, v=<2,3> and w=<3,2>. Determine if u,v, and w are linear independent.
[1 2 3, 4 3 2| 0, 0]
R2: R2-4R1
[1 2 3 , 0 -5 -10| 0, 0]
R2: 1/5(R2)
[1 2 3, 0 -1 -2|0, 0]
R1: R1-2R2
[1 0 -1, 0 -1 -2| 0 0]
Create system of equations and solve:
a-c b+2c
Subtracting “b” from both sides on the second equation:
b=-2c
Conclusion: This shows that the vectors are linearly dependent because there is a nontrivial linear combination that equals to 0.
1.For any transformation matrix such that T=Ax , if A is invertible then T is invertible.
This is true. If A is invertible then, T is invertible.
2.Provide an example of two vectors that form the basis of R^2.
<1,0> and <0,1> form a basis of R^2.