Linear Algebra Concepts

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the null space of matrix A?

The null space of a matrix A is the set Nul A of all solutions of the homogeneous equation Ax=0. The null space of an m × n matrix A is a subspace of Rⁿ.

What are the properties of a subspace H in Rⁿ? (Select all that apply)

  • For each u and v in H, the sum u + v is in H (correct)
  • For each u in H and each scalar c, the vector cu is in H (correct)
  • The zero vector is in H (correct)
  • H contains only vectors of the same length

What is the column space of matrix A?

The column space of a matrix A is the set Col A of all linear combinations of the columns of A.

The set {e₁;...; eη} is called the __ for Rⁿ.

<p>standard basis</p> Signup and view all the answers

True or false: The pivot columns of a matrix A form a basis for the column space of A.

<p>True (A)</p> Signup and view all the answers

What is the Diagonalization Theorem?

<p>An nxn matrix A is diagonalizable if and only if A has n linearly independent eigenvectors.</p> Signup and view all the answers

Define the dimension of a nonzero subspace H.

<p>The dimension of a nonzero subspace H is the number of vectors in any basis for H.</p> Signup and view all the answers

True or false: The rank of a matrix A, denoted by rank A, is the dimension of the column space of A.

<p>True (A)</p> Signup and view all the answers

What is the rank for matrix A?

<p>The rank of a matrix A, denoted by rank A, is the dimension of the column space of A.</p> Signup and view all the answers

What is the rank theorem?

<p>If a matrix A has n columns, then rank A + dim Nul A = n.</p> Signup and view all the answers

What is the basis theorem?

<p>Let H be a p-dimensional subspace of Rⁿ. Any linearly independent set of exactly p elements in H is automatically a basis for H.</p> Signup and view all the answers

What are the equivalent statements to the invertible theorem for a matrix A? (Select all that apply)

<p>dim Col A = n (A), The columns of A form a basis of Rⁿ (B), rank A = n (C), Col A = Rⁿ (D)</p> Signup and view all the answers

What is an eigenvector of a matrix?

<p>An eigenvector of an n × n matrix A is a nonzero vector x such that Ax = λx for some scalar λ.</p> Signup and view all the answers

What is called an eigenvalue of A?

<p>A scalar.</p> Signup and view all the answers

True or false: λ is an eigenvalue of an n × n matrix A if and only if the equation (A - λI)x = 0 has a nontrivial solution.

<p>True (A)</p> Signup and view all the answers

True or false: The eigenvalues of a triangular matrix are the entries on its main diagonal.

<p>True (A)</p> Signup and view all the answers

What does it mean if the number 0 is not an eigenvalue of A?

<p>The matrix A is invertible (D)</p> Signup and view all the answers

Properties of Determinants: A is invertible if and only if ___

<p>det A ≠ 0 (A)</p> Signup and view all the answers

What is the characteristic equation for an n × n matrix A?

<p>det(A-λI) = 0</p> Signup and view all the answers

True or false: If n × n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues.

<p>True (A)</p> Signup and view all the answers

True or false: An n × n matrix with n distinct eigenvalues is not diagonalizable.

<p>False (B)</p> Signup and view all the answers

Let A be an n × n matrix whose distinct eigenvalues are λ₁;...; λₚ. The dimension of the eigenspace for λκ is ______.

<p>the number of linearly independent eigenvectors corresponding to λκ.</p> Signup and view all the answers

Flashcards are hidden until you start studying

Study Notes

  • Null space (Nul A) is the set of all solutions to the homogeneous equation Ax=0.
  • Nul A is a subspace of Rⁿ, defined as Nul(A) = {x | Ax=0}, where x is in Rⁿ.
  • Null space theorem is important in understanding the solution space of linear equations.

Subspaces

  • A subspace H in Rⁿ must contain the zero vector, support closure under addition, and allow scalar multiplication.
  • For each u and v in H, u + v must also be in H; for each u in H and scalar c, cu must be in H.

Column Space

  • Column space (Col A) consists of all linear combinations of the columns of matrix A.

Standard Basis

  • The set {e₁, ..., eη} represents the standard basis for Rⁿ, providing a fundamental set of vectors.

Pivot Columns

  • The pivot columns of a matrix A form a basis for its column space, confirming their significance in linear independence.

Diagonalization Theorem

  • An nxn matrix A is diagonalizable if it has n linearly independent eigenvectors.
  • This can be represented as A = PDP⁻¹, where D is a diagonal matrix containing eigenvalues corresponding to eigenvectors in P.

Dimension of Subspaces

  • The dimension of a nonzero subspace H is determined by the number of vectors in any basis for H.
  • The dimension of the zero subspace {0} is defined to be zero.

Rank of a Matrix

  • The rank of matrix A is the dimension of its column space, denoted as rank A.
  • Rank theorem states that for an n-column matrix A, rank A + dim Nul A = n.

Basis Theorem

  • A linearly independent set of exactly p elements in a p-dimensional subspace H is a basis for H.
  • Similarly, any set of p elements that spans H is also a basis for H.

Invertible Matrix Theorem

  • An n × n matrix A is invertible if its columns form a basis of Rⁿ, satisfying various conditions related to rank and null space.

Eigenvectors and Eigenvalues

  • An eigenvector of matrix A satisfies the equation Ax = λx for some scalar λ.
  • An eigenvalue corresponds to a nontrivial solution of Ax = λx.

Characteristic Equation

  • A scalar λ is an eigenvalue of matrix A if it satisfies the equation det(A - λI) = 0.

Properties of Triangular Matrices

  • The eigenvalues of a triangular matrix are the entries on its main diagonal.

Determinants

  • A matrix A is invertible if and only if det A ≠ 0.
  • Properties include det(AB) = (det A)(det B) and det Aⁿ = det A raised to the power of n.
  • Row operations affect determinants: row replacement maintains it, row interchange changes the sign, and row scaling multiplies by the scalar.

Similar Matrices

  • Similar n × n matrices have the same characteristic polynomial and eigenvalues, ensuring they share essential properties.

Diagonalizability

  • An n × n matrix with n distinct eigenvalues is guaranteed to be diagonalizable.
  • The dimension of the eigenspace for eigenvalues plays a crucial role in determining diagonalizability.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Quiz Team

More Like This

Linear Algebra: Range and Null Space
6 questions
Linear Algebra: Null Space and Column Space
9 questions
Linear Algebra Chapter 4 T/F Quiz
23 questions

Linear Algebra Chapter 4 T/F Quiz

BeneficialThermodynamics avatar
BeneficialThermodynamics
Use Quizgecko on...
Browser
Browser