Linear Equations Systems

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Questions and Answers

What can be concluded if a system of 2 linear equations in 3 unknowns has exactly one solution?

The equations must be dependent.

The system is inconsistent.

The equations are independent. (correct)

There are infinitely many solutions.

Which statement is true about a system of linear equations that has a pivot in every row of matrix A?

The system is inconsistent for every b.

The system may be inconsistent for some values of b.

The system must have infinitely many solutions.

The system is consistent for every b. (correct)

If the augmented matrix has a pivot in the last column, what can be said about the system Ax = b?

The system has a unique solution.

The system has multiple solutions.

The system is inconsistent. (correct)

The system is consistent.

Which of the following confirms that a transformation T is linear?

T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all u, v, and c. (B) Signup and view all the answers

What is the requirement for matrix A to be invertible considering its null space?

Null(A) = {0} implies A is invertible. (B) Signup and view all the answers

In which scenario is the product AB of two matrices guaranteed to be invertible?

If both A and B are invertible. (B) Signup and view all the answers

Which of the following is true about a row of zeros in matrix A?

It implies that the system can still have a solution depending on b. (B) Signup and view all the answers

Given that {u, v, w} is linearly dependent, what can be stated about {Au, Av, Aw}?

It is guaranteed to be linearly dependent. (D) Signup and view all the answers

If a matrix A is row-equivalent to the identity matrix, what can be concluded about A?

A is a square matrix. (A), Ax = 0 implies x = 0. (B) Signup and view all the answers

Which statement about the determinant of a matrix A is correct?

If det(A) = 0, then A is not invertible. (A) Signup and view all the answers

Which of the following statements regarding subspaces is true?

A subspace W must contain the zero vector. (B) Signup and view all the answers

If a matrix A has n pivot columns, what can be concluded about its null space?

Nul(A) is {0}. (B) Signup and view all the answers

For a fixed vector b ≠ 0, how can the set of solutions to Ax = b be characterized?

It is not a subspace since it does not include the zero vector. (A) Signup and view all the answers

If det(A) = 1 and A consists of integer entries, what can be said about A's inverse?

A−1 must have integer entries. (D) Signup and view all the answers

Which statement about vector spaces is correct?

A set must be closed under addition and scalar multiplication to be a vector space. (D) Signup and view all the answers

What happens when a matrix A is multiplied by a scalar 2?

The determinant doubles: det(2A) = 2 det(A). (A) Signup and view all the answers

Which statement is true regarding row operations on a matrix?

Row operations preserve the linear independence relations of the columns of a matrix. (A) Signup and view all the answers

If B is a spanning subset of an n-dimensional vector space V, what can be said about B?

B is not necessarily a basis for V. (B) Signup and view all the answers

Which of the following statements is correct regarding eigenvalues and diagonalizability?

A matrix is diagonalizable if it has enough linearly independent eigenvectors. (D) Signup and view all the answers

What does it imply if 0 is an eigenvalue of matrix A?

Matrix A has zero determinant. (C) Signup and view all the answers

What can be concluded about the projections in a subspace W?

The orthogonal projection of any vector onto W is the vector itself. (D) Signup and view all the answers

If A is similar to B, which of the following statements is true?

A and B have the same eigenvalues. (A) Signup and view all the answers

Which condition affects the invertibility of a matrix?

If all eigenvalues of a matrix are non-zero, it is invertible. (D) Signup and view all the answers

What results from the characteristic polynomial of A being given as λ² - 3λ + 2 = 0?

A² - 3A + 2I = 0, where I is the identity matrix. (B), A has eigenvalues 1 and 2. (D) Signup and view all the answers

Study Notes

Chapter 1: Systems of Linear Equations

A system of 3 linear equations in 2 unknowns must have no solution
A system of 2 linear equations in 3 unknowns could have exactly one solution
A system of linear equations cannot have exactly two solutions
If there's a pivot in every row of A, then Ax = b is consistent for all b
If the augmented matrix has a pivot in the last column, then Ax = b is inconsistent
If A has a row of zeros, then Ax = b is inconsistent for all b
Ax = 0 is always consistent
If {u, v, w} is linearly dependent, then {Au, Av, Aw} is also linearly dependent for every A
If {u, v, w} is linearly independent, and {v, w, p} is linearly independent, then {u, v, w, p} is also linearly independent
If {u, v, w} is linearly dependent, then u is in the span of {v, w}
If {u, v, w} is linearly dependent and {u, v} is linearly independent, then w is in the span of {u, v}
A linear transformation from R² to R³ has a 2 × 3 matrix

Chapter 2: Matrix Algebra

AB + BT – AT is always symmetric
Any matrix can be written as a sum of a symmetric and antisymmetric matrix.
(AB)⁻¹ = A⁻¹B⁻¹
If AB = AC, then B = C
The matrix [1 2 3] / [3 6 9] is not invertible
If AB = I for some B, then A is invertible
A 3 × 2 matrix could be invertible
A 2 × 3 matrix could be invertible
If AB is invertible, then A and B are invertible (if A and B are square)
If Nul(A) = {0}, then A is invertible

Chapter 3: Determinants

In general, det(2A) = 2ⁿ det(A) where n is the dimension of the matrix.
det(A + B) ≠ det(A) + det(B)
If det(A²) + 2 det(A) + det(I) = 0, then A is invertible
det(A⁻¹) = 1/det(A)
If A¹⁰⁰ is invertible, then A is invertible

Chapter 4: Vector Spaces and Subspaces

{(x, y) ∈ R² | x² + y² = 0} is a subspace of R²
The union of two subspaces of V is not always a subspace of V
The intersection of two subspaces of V is a subspace of V
Given any basis B of V, and a subspace W of V, then there is a subset of B that is a basis of W.
R² is a subspace of R³

Chapter 5: Eigenvalues and Eigenvectors

A 3 × 3 matrix with eigenvalues λ = 1, 2, 4 must be diagonalizable
A 3 × 3 matrix with eigenvalues λ = 1, 1, 2 is not diagonalizable
Every matrix is not necessarily diagonalizable
If A is similar to B, then det(A) = det(B)
If A is similar to B, then A and B have the same eigenvalues
If A is diagonalizable, then det(A) is the product of the eigenvalues of A
If A is similar to B, then A and B have the same eigenvectors

Chapter 6: Orthogonality and Least-Squares

If x is the orthogonal projection of x on a subspace W, then x is perpendicular to x - x
x = x
The orthogonal projection of x on W⁺ is x - x
Every (nonzero) subspace W has an orthonormal basis
W∩W⁺ = {0}
AATx is the projection of x on Col(A)
Same, but the columns of A are orthonormal
Rank(ATA) = Rank(A)
If Q is an orthogonal matrix, then Q is invertible

Chapter 7: Symmetric Matrices

If A is symmetric, then eigenvectors corresponding to different eigenvalues are orthogonal
A symmetric matrix has only real eigenvalues
Linearly independent eigenvectors of a symmetric matrix are orthogonal
If A is symmetric, then A is orthogonally diagonalizable
If A is orthogonally diagonalizable, then A is symmetric

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This quiz covers key concepts from Chapter 1 on systems of linear equations, including the conditions for consistency and dependence among equations. Test your understanding of how multiple equations interact and the implications of pivots in matrices.

Linear Equations Systems - Chapter 1

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