Word Problems and Concepts
A system of equations that has zero solutions is described by this single word.
Topic: Solving Linear Equations
Answer: Inconsistent
The leftmost nonzero entry in a row of a matrix in echelon form.
Topic: Echelon vs RREF
Answer: Leading entry (or Pivot)
Topic: Determinants (2x2)
Answer: 5(1) - 2(3) = -1
Topic: Linear Independence
Answer: Yes
If a 3x5 matrix A is row-reduced and has 3 pivot positions, what is the dimension of the Column Space?
Topic: Rank and Nullity
Answer: 3 (the rank(
The term for a solution where all the weights c_i in a linear combination are zero.
Topic: Trivial vs Non-Trivial Solution
Answer: Trivial solution
Find the dot product of u = [1 2 3] and v = [0 1 -1]
Topic: Dot Product
Answer: 2 + (-3) = -1
Given the eigenvalue λ = 2, find the corresponding eigenvector for the matrix [3 0; 1 2].
Topic: Eigenvectors
Answer: Solve (A - 2I)x = 0. x = [0 1] or any multiple of that vector
The set of vectors that forms the basis for a vector space must satisfy which two key properties?
Topic: Vector Space
Answer: Spans the space AND is linear independent
Two vectors, u and v, are said to be this if their dot product equals zero.
Topic: Orthogonality
Answer: Orthogonal
Find the magnitude of the vector v = [3; -4]
Topic: Magnitude of Vector
Answer: ||v|| = sqrt (3^2 + (-4)^2) = 5
Answer: 200
True or False: If a square matrix A has a non-zero determinant, then the homogeneous system Ax = 0 must have only the trivial solution.
Topic: Determinants
Answer: True (by the Invertible Matrix Theorem)
The non-zero vector x that satisfies the equation Ax = λx.
Topic: Eigenvectors
Answer: Eigenvector
If A is a matrix whose columns are dependent, what is det(A)?
Topic: Linear Independence
Answer: 0
What is the dimension of the Null Space (Nullity) of the matrix A in RREF [1 0 3 0; 0 1 1 0; 0 0 0 1]?
Answer: 1 (because there is 1 free variable x3)
If T(x) = Ax and the transformation maps R3 onto R4, what must be true about the rank of the 4x3 matrix A?
Topic: Linear Transformations
Answer: Rank must be 4 (This is impossible, so the map is not onto R4)
This theorem links a matrix's rank, nullity, and the number of columns. What does it state?
Topic: Rank and Nullity
Answer: Rank + Nullity = n
For A = [2 1; 3 4], find the transpose of matrix A
Topic: Matrix Transpose
Answer: [2 3; 1 4]
Given basis B = {[1 0], [1 1]} and basis C = {[1 2], [2 1]} and [x]B = [3 2], what is [x]C?
Topic: Change of Coordinate
Answer: [x]C = [-1/3 8/3]