Linear Equations Systems - Chapter 1

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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?

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

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

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

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

<p>If both A and B are invertible. (B)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

Which of the following statements regarding subspaces is true?

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

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

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

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

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

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

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

Which statement about vector spaces is correct?

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

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

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

Which statement is true regarding row operations on a matrix?

<p>Row operations preserve the linear independence relations of the columns of a matrix. (A)</p> 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?

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

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

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

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

<p>Matrix A has zero determinant. (C)</p> Signup and view all the answers

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

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

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

<p>A and B have the same eigenvalues. (A)</p> Signup and view all the answers

Which condition affects the invertibility of a matrix?

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

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

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

Flashcards

Row operations preserve column linear independence

Row operations on a matrix do not change the linear independence relationships between its columns.

Spanning set to basis

Any set of vectors that spans a vector space can be reduced to a linearly independent subset that forms a basis for the space.

Linearly independent set in n-dimensional space

If a set of vectors is linearly independent and has the same number of vectors as the dimension of the vector space, then it forms a basis.

Spanning set in n-dimensional space

If a set of vectors spans an n-dimensional vector space and has the same number of vectors as the dimension of the vector space, then it forms a basis.

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Dimension of P4

The dimension of the vector space of polynomials of degree at most 4 (P4) is 5, not 4.

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Matrix P and basis B

If B is a basis for Rn, and matrix P has the vectors of B as its columns, then multiplying P by a vector x gives the coordinates of x with respect to basis B.

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Change of basis matrix

Given bases B and C for Rn, and matrix P formed by the coordinate vectors of B with respect to C, then multiplying P by the coordinates of x with respect to C produces the coordinates of x with respect to B.

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Coordinates with respect to standard basis

The standard basis for Rn is a set of vectors where each vector has a single 1 and the rest are 0s. The coordinates of a vector x with respect to the standard basis are simply the components of x.

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System of 3 equations, 2 unknowns

A system with more equations than unknowns (e.g., 3 equations, 2 unknowns) doesn't always have a solution. Think of a system trying to satisfy more conditions than it has variables to adjust.

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System of 2 equations, 3 unknowns

A system with more unknowns than equations (e.g., 2 equations, 3 unknowns) could have a single solution. The equations might be independent and define a point where all variables are consistent.

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Linear system with exactly two solutions

Linear equations are lines or planes. They can intersect in at most a single point, or infinitely many points if they coincide. Therefore, they cannot intersect at two specific points.

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Pivot in every row of A

If every row of the augmented matrix has a pivot, there is a leading '1' in every row. This indicates a unique solution for each variable. Every equation has a unique solution, making the system consistent.

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Pivot in last column of augmented matrix

If the augmented matrix has a pivot in the last column, it represents a row with all zeros except for a '1' in the last column. This indicates an inconsistent equation, meaning no solution satisfies the system.

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Matrix A with a row of zeros

A row of zeros means a dependent equation, not a contradiction. The system might be consistent, with infinitely many solutions. We can't determine inconsistency solely based on a zero row.

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Ax = 0 is always consistent

The equation Ax = 0 always has the trivial solution x = 0. The system is always consistent because setting all variables to zero satisfies the equations.

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Linear dependence under transformation

Linear dependence implies that one vector can be expressed as a combination of others. Applying a linear transformation (A) preserves this relationship, making {Au, Av, Aw} also linearly dependent.

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One-to-one linear transformations are onto

If a linear transformation from Rn to Rn is one-to-one, it means that every vector in the codomain (Rn) is the image of exactly one vector in the domain (Rn). This implies that the transformation maps the entire domain onto the entire codomain.

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Row operations and inverses

The row operations that transform a square matrix (A) into the identity matrix (I) are the same operations that transform the identity matrix (I) into the inverse of the original matrix (A^-1).

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Row equivalence and homogeneous systems

A square matrix (A) is row-equivalent to the identity matrix if and only if the only solution to the homogeneous system Ax = 0 is the trivial solution (x = 0).

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Determinant scaling

In general, the determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the matrix's dimension multiplied by the determinant of the original matrix.

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Determinant of a sum

The determinant of the sum of two matrices is not equal to the sum of their determinants, unless one of the matrices is a zero matrix.

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Invertibility equation

If the scalar multiplication by a matrix on itself (squared) plus twice the determinant of the matrix itself plus the determinant of the identity matrix equals zero, then the matrix is invertible.

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Determinant of inverse

The determinant of the inverse of a matrix is equal to the reciprocal of the determinant of the original matrix.

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Power and invertibility

If a matrix raised to a power (100 in this case) is invertible, then the original matrix must also be invertible.

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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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