Comment due Sunday, March 25
Test #2 will be given in class Monday, April 1. The format will be similar to the format of Test #1.
Recall from Test #1 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 #2, 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 questions for inspiration (the different versions of the tests had similar questions, so check out your classmates’ solutions for the other versions).
Try to focus on the material covered in class since Test #1. You can see the list of topics on the schedule.
1.Statement: “Cramer’s Rule can be applied to any square matrix.”
Answer: False
Explanation: Cramer’s Rule can only be applied to square matrices that are non-singular, meaning their determinant is non-zero. If the determinant is zero, the matrix is singular, and Cramer’s Rule cannot be used.
2.Example:
Question: Provide an example of two vectors in R^3 that are orthogonal to each other.
Answer: Two vectors that are orthogonal to each other in R^3 are v = ⟨1,0,0⟩ and w =⟨0,1,0⟩. These vectors are perpendicular because their dot product is zero: v ⋅ w =1⋅0+0⋅1+0⋅0=0.
1)”Assume {u1,u2,u3,u4} does not span R^3. Is it true that {u1,u2,u3} also cannot span R^3?”
It is true because removing a vector from the set can only decrease the span and cannot increase it.
2) “Can you show a linear equation of the plane through the origin and perpendicular to the vector <-5,-1,-1>?
The equation would be -5x-y-z=0 because you just multiply XYZ with their corresponding number and perpendicular means it must be equal to zero.
false. the linear system Ax=0 is always consistent regardless of the matrix A.
2.what is required for a set of Cols to span R^n
if there’s a pivot in every row so a matrix W/3 rows, and 3 pivots spans R^3
1.- If vectors u and v are in the x y plane then u x v points in the direction of the y axis.
False, the cross product of to vectors in the x y plane points in the direction of the z axis
2.- Provide a vector n that is normal to a plane with vector <2, 5>.
vector n = <-5, 2> is normal to the plane where vector <2, 5> is found as n * v = 0
False. Matrix, which determinant’s is zero, doesn’t have inverse.
2 What does it mean that two vectors are orthogonal?
It means that they are perpendicular to each-other and their dot product is zero.
(1) Cramer’s rule can be used on Ax=b when det(A)=0.
False, because when det(A)=0, x=det(A1)/det(A) is undefined.
(2) Provide an example of two parallel victors in an R^2 space.
<1,1> and <-1, -1>, because A=kB => <1, 1>=-1<-1, -1>.
Statement 1: The norm of a vector is always non-negative.
Answer: True
explanation: Norms measure vector length, always yielding non-negative values.
Statement 2: The distance between two vectors is the same as the norm of their sum.
Answer: False
explanation: Distance between vectors equals the norm of their difference, not their sum.
Q1: Interchanging two rows of an invertible matrix does not change its determinant (True/False)
A1: False, det(b) = -det(a) if two rows of an invertible matrix swap
Q2: Provide a linear (scalar) equation of a plane in R^2 along with a coordinate within the plane
A2:
Equation: 4x+5y = 0
Coordinates: (0,0)
1) Let a1 = [8,-3,3] , a2 = [-1,-8,8], and b = [-35,-12,12] , B is not in the Span?
False, B is in the Span (Vector b is a linear combination of a1and a2 proven by solving the system of linear equations.)
2.) if rref(A) = Rredf(B), then Row(A) = Row (B)
true ( If the reduced row echelon forms (RREF) of matrices A and B are the same, then the row spaces of A and B are identical. This is because RREF is a unique matrix representation that directly reflects the linear dependencies among the rows, thus defining the row space)
Expaination:
Question 1
Statement: “If the determinant of a matrix A is zero, then A is invertible.”
Answer: False
Explanation: If the determinant of a matrix A is zero, the matrix is singular and does not have an inverse. A matrix is invertible only if its determinant is non-zero.
Question 2
Question: Provide an example of a 3×3 matrix that is not invertible