18 Questions
What is the solution to the system of linear equations £1—Xo+ x3 =0, 221 —X2 =2, and —32} +223 =1?
x1 = -5, x2 = -12, x3 = -7
What is the value of h such that the matrix is the augmented matrix of a consistent linear system?
h = 3/2
What is the condition for the vector b to be in the range of the linear transformation T(x) = Ax?
The vector b is a linear combination of the columns of A.
What is the solution to the vector equation 2x + 2y - 3z = 1, x - 2y + z = -3, and x + y - z = 2?
x = 3/2, y = -1/2
Suppose T : R² → R³ is a linear transformation. If T(u) = [1, 2, 3] and T(v) = [4, 5, 6], what is T(2u - v)?
[-2, -3, -4]
What is the standard matrix of the linear transformation T that maps e₁ to e₁ + e₂ and e₂ to e₂ - 2e₁?
[[1, 1], [1, -2]]
What is the description of the entire solution set to Ax = 0?
x = t[1] + s[-1] + r[1]
Is the set of vectors { [|1], [|2], [|3] } linearly independent?
linearly dependent
Is the linear transformation T(x) = [| -2x₁ + x₂, x₁ + 3x₂ |] one-to-one and/or onto?
It is one-to-one but not onto.
Let T : R³ → R² be a linear transformation with standard matrix A. If Ax = b has a solution for every b in R², what can be said about the matrix A?
The matrix A is onto.
What is the inverse of the matrix A if the solution set to Ax = b is described as x = t[1] + 4s[1] + r[-1]?
A^(-1) = [1 4 -1]
Let T : R² → R² be a linear transformation with standard matrix A. If T is invertible, what can be said about the matrix A?
The matrix A is invertible.
If an m x n matrix has m pivot columns, what can be said about the linear transformation T(x) = Ax?
It is a one-to-one mapping.
If A is an m x n matrix and the equation Ax = b is consistent for some b, what can be concluded about the columns of A?
The columns of A span R^n.
If an n x n matrix has n pivot positions, what can be said about the reduced echelon form of A?
The reduced echelon form of A is an identity matrix.
If A and B are m x n matrices, what can be said about the matrices AB^T and A^TB?
Both AB^T and A^TB are defined.
If A is invertible and r ≠ 0, what can be said about the inverse of (rA)?
The inverse of (rA) is (1/r)A^(-1).
If AB = AC where B and C are n x p matrices, and A is an invertible n x n matrix, what can be concluded about B and C?
B = C.
This quiz covers linear algebra concepts, including linear independence, dependence, and linear transformations. It includes multiple-choice questions and a problem-solving question on finding the values of x that satisfy a given linear transformation.
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