Comment due Sunday, April 29
Test #3 will be given in class Monday, May 6. 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.
1. Vectors in a orthogonal set are always linearly independent.
True, because vectors in a orthogonal set are perpenticular to each other, so they are linearly independent.
2. Give one of the standard bsis of IR^3.
<1,0,0>, answers can be either <1,0,0>, <0,1,0>, or <0,0,1>.
1) A subspace of R^n must contain the zero vector. True because it needs to have closure under addition and scalar multiplication.
2)Give an example of a non-empty subset of R^2 that is not a subspace. <2,4> and <2,2> are a subset of R^2 but not a subspace because it does not contain a zero vector
True/False: If nxn matrix A is invertible, then its columns are linearly independent.
True, because for a matrix A to be invertible, there exists a matrix A(-1). To be invertible for the columns the equation Ax=0 must be true. Then A(-1)*A*x=0 which is also x=0. Therefore the columns of A is invertible.
Give an example of 3*5 matrix with rank 2
<2,1,2>,<2,1,2>,<4,5,2>,<2,1,2>,<2,4,1> because there are three columns that are equal to each other and two other columns are linearly independent, rank of the matrix is 2
Question 1
Statement: “The row space of a matrix is equal to the column space of its transpose.”
Answer: True
Explanation: The row space of a matrix A consists of all linear combinations of its rows, while the column space of the transpose of A (i.e., A^T) consists of all linear combinations of the columns of A^T, which are the rows of A. Therefore, the row space of A is indeed equal to the column space of A^T.
Question 2
Question: Provide an example of a 2×2 matrix that is diagonalizable but not invertible.
1.- If A is nxn, then A is invertible if and only if rank(A) < n.
False, A is invertible if and only if it is full rank, meaning rank(A) = n
2.- Provide a linearly dependent subset that spans R2:
{(1,0),(0,1),(1,1)} They can be written as a linear combination of the others.