Linear Algebra Concepts Quiz

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Questions and Answers

What is the rank of a matrix A?

The number of non-zero rows in any row-echelon form of A (correct)

The number of columns in A

The number of zero rows in any row-echelon form of A

The total number of rows in A

What is a linear combination of vectors?

A vector v is a linear combination of vectors v1, v2,...,vR if there are scalars c1, c2,..., cR such that v = c1v1 + c2v2 +...+ cRvR.

A set of vectors is linearly dependent if all scalars are zero.

False

Define the inverse of a matrix.

A matrix A' is an inverse of A if A'A = AA' = I. Signup and view all the answers

An elementary matrix is obtained from an identity matrix by performing one ______ operation.

row Signup and view all the answers

What are the conditions for a collection of vectors to be a subspace?

<ol> <li>0 E S, 2. if u, v E S, then u + v E S, 3. if u E S, c E R then cu E S.</li> </ol> Signup and view all the answers

What is a basis for a subspace S of Rn?

A set of vectors in S that spans S and is linearly independent. Signup and view all the answers

What defines the dimension of a subspace?

The number of vectors in a basis for the subspace. Signup and view all the answers

Define a linear transformation.

A mapping T: Rn - Rm is a linear transformation if for each c1, c2 E R and v1, v2 E Rn, T(c1v1 + c2v2) = c1T(v1) + c2T(v2). Signup and view all the answers

What is an eigenvalue?

A scalar (lambda) is called an eigenvalue of A if there is a non-zero vector x such that Ax = (lambda)x. Signup and view all the answers

What does it mean for two matrices to be similar?

A is similar to B if there is an invertible matrix P such that P^-1AP = B. Signup and view all the answers

What is a diagonalizable matrix?

An nxn matrix A is diagonalizable if there exists a diagonal matrix D such that A is similar to D. Signup and view all the answers

Define the null space of a matrix.

The null space of A is the subspace of Rn consisting of solutions to the homogeneous linear system Ax = 0. Signup and view all the answers

What is the row space of a matrix?

The row space of A is the subspace spanned by the rows of A. Signup and view all the answers

Define the column space of a matrix.

The column space of A is the subspace spanned by the columns of A. Signup and view all the answers

What is the nullity of a matrix?

The nullity of a matrix A is the dimension of its null space. Signup and view all the answers

What is the standard matrix of a linear transformation?

T = T_A where A (mxn) is the standard matrix A = [T(e1):T(e2):...:T(en)]. Signup and view all the answers

What is the characteristic polynomial?

The polynomial obtained by expanding det(A - lambdaI) to determine eigenvalues. Signup and view all the answers

Define the characteristic equation.

The equation det(A - lambdaI) = 0. Signup and view all the answers

What is the eigenspace of an eigenvalue?

The collection of all eigenvectors corresponding to lambda, together with the zero vector. Signup and view all the answers

What is algebraic multiplicity?

The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic equation. Signup and view all the answers

Define geometric multiplicity.

The geometric multiplicity of an eigenvalue is the dimension of its corresponding eigenspace. Signup and view all the answers

What does the Basis Theorem state?

Any two bases for a subspace have the same number of vectors. Signup and view all the answers

What does the Rank Theorem state?

If A is an mxn matrix, then rank(A) + nullity(A) = n. Signup and view all the answers

What is the Fundamental Theorem of Invertible Matrices?

The statements equivalent to A being invertible include unique solutions to Ax = b for every b in Rn. Signup and view all the answers

Study Notes

Rank of a Matrix

Rank represents the number of non-zero rows in any row-echelon form of matrix A.
Denoted as rank(A).

Linear Combination of Vectors

A vector v can be expressed as a linear combination of vectors v1, v2,..., vR using scalars c1, c2,..., cR such that v = c1v1 + c2v2 +...+ cRvR.

Linear Dependence

A set of vectors v1, v2,..., vR in Rn is linearly dependent if there exist scalars c1, c2,..., cR, not all zero, such that c1v1 + c2v2 +...+ cRvR = 0.
If no such scalars exist, the set is independent.

Inverse of a Matrix

An nxn matrix A has an inverse A' if A'A = AA' = I, where I is the identity matrix.

Elementary Matrices

Elementary matrices are formed from the identity matrix by performing a single row operation.

Subspace

A subspace of Rn must satisfy:
- Includes the zero vector (0 ∈ S).
- Closed under addition (if u, v ∈ S, then u + v ∈ S).
- Closed under scalar multiplication (if u ∈ S and c ∈ R, then cu ∈ S).

Basis

A basis for a subspace S of Rn consists of vectors that both span S and are linearly independent.

Dimension of a Subspace

The dimension of a subspace S is determined by the number of vectors in a basis for that subspace.

Linear Transformation

A mapping T: Rn → Rm is a linear transformation if it satisfies T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for any scalars c1, c2 and vectors v1, v2.

Eigenvalues and Eigenvectors

For an nxn matrix A, a scalar λ is an eigenvalue if there exists a non-zero vector x such that Ax = λx.
Such a vector x is called an eigenvector corresponding to λ.

Similar Matrices

Matrices A and B are similar (A ~ B) if there exists an invertible matrix P such that P⁻¹AP = B.

Diagonalizable Matrices

An nxn matrix A is diagonalizable if there is a diagonal matrix D such that A is similar to D via an invertible nxn matrix P (P⁻¹AP = D).

Null Space

The null space of an mxn matrix A consists of all solutions to the homogeneous equation Ax = 0, denoted as null(A).

Row Space

The row space of an mxn matrix A is the subspace of Rn spanned by the rows of A.

Column Space

The column space of an mxn matrix A is the subspace of Rm spanned by the columns of A.

Nullity of a Matrix

Nullity is the dimension of the null space of matrix A, denoted as nullity(A).

Standard Matrix of a Linear Transformation

For a linear transformation T: Rn → Rm, the standard matrix A can be represented as A = [T(e1), T(e2), ..., T(en)], where e1, e2, ..., en denote the standard basis vectors.

Characteristic Polynomial

The characteristic polynomial is found by expanding det(A - λI) to determine eigenvalues.

Characteristic Equation

The characteristic equation is expressed as det(A - λI) = 0.

Eigenspace

The eigenspace corresponding to an eigenvalue λ contains all eigenvectors associated with λ, including the zero vector, denoted as E(λ).

Algebraic Multiplicity

The algebraic multiplicity of an eigenvalue refers to its occurrence as a root in the characteristic equation.

Geometric Multiplicity

The geometric multiplicity is defined as the dimension of the eigenspace corresponding to an eigenvalue.

The Basis Theorem

Any two bases for a subspace S have the same number of vectors.

The Rank Theorem

For an mxn matrix A, the relationship rank(A) + nullity(A) = n holds true.

The Fundamental Theorem of Invertible Matrices

An nxn matrix A is invertible if and only if:
- Ax = b has a unique solution for every b in Rn.
- Ax = 0 has only the trivial solution.
- The reduced row echelon form of A equals the identity matrix I.
- A can be expressed as a product of elementary matrices.
- rank(A) = n.
- nullity(A) = 0.

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Description

Test your understanding of key concepts in Linear Algebra, including matrix rank, linear combinations, linear dependence, and the properties of matrices. This quiz will help solidify your grasp of foundational topics essential for advanced studies in mathematics.