Linear Algebra Exam 3 T/F and Practice Questions

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a set B={v1,v2,...,vp} of vectors is said to be a BASIS for a subspace H of R^n when...

(1) H contains {v1,v2,...,vp}, (2) H = Span{v1,v2,...,vp}, (3) {v1,v2,...,vp} is linearly independent.

Some other simple criteria for linear independence we met earlier will be also be useful:

(1) a set {v} containing only one non-zero vector is linearly independent because x1v=0 only if x1=0, (2) a set {v1,v2} containing two vectors is linearly dependent if and only if v2 is a scalar multiple of v1 because x1v1+x2v2=0 → v1= −(x2/x1)v2 or v2= −(x1/x2)v1 unless x1=x2=0, (3) any set {v1,...,vk−1,0,vk+1,...,vp} containing the zero vector is linearly dependent because: c1v1+...+ck−1vk−1+ck0+ck+1vk+1+⋯+cpvp = 0 when cj=0 for j≠k but ck=1.

basic theorem: for an m×n matrix A:

1) the columns of a matrix A corresponding to the pivot columns of rref[A 0] form a basis for Col(A), 2) the dimension of Nul(A) is the number of free variables in the equation Ax=0, 3) the dimension of Col(A) is the number of basic variables in the equation Ax=0.

Basis

A basis for a vector space is a sequence of vectors that form a set that is linearly independent and that spans the space.

A matrix is invertible when

A square matrix is singular if and only if its determinant is 0. So a matrix is invertible when the determinant is 0.

Definition of Eigenvalue and Eigenvector

Definition: a scalar λ is said to be an EIGENVALUE of an n×n matrix A if there exists a non-trivial solution x in R^n of the equation Ax=λx; such an x is said to be an EIGENVECTOR corresponding to λ.

v1 and v2 are linearly independent eigenvectors of an nxn matrix A then they correspond to distinct eigenvalues of A. True or False?

FALSE!

Three vectors, one of which is the zero vector, make a basis for R^3. True or false?

False! No set of vectors containing the zero vector can be linearly independent. Thus no set of three vectors containing the zero vector can be a basis for R^3.

To find the eigenvalues of A, reduce A to echleon form. True or False?

False.

find a basis for R^n

Find the associated augmented matrix for Ax=0 and then you use the # of pivot columns of rref(A0) to be the basis of the Col(A) or the # of free variables to be the basis of Nul(A)

When b1, b2, . . . , bp are vectors in R^n and H = Span{b1, b2, . . . , bp}, then {b1, b2, . . . , bp} is a basis for H. True or False?

For the set {b1, b2, . . . , bp} to be a basis for H, BOTH of the conditions (i) H = Span{b1, b2, . . . , bp} , (ii) the set is linearly independent, must be met. so the condition is false.

Find the basis set of a matrix A

Just find the rref (A0) and the pivot columns revel the basis set of A to be {a1,..}

The dimensions of the column space of A is equal to the rank of A. True or False

TRUE!

Col (A) is ...?

The column space of an mxn matrix A (Col A) is the set of all linear combinations of the columns of A. If A = {a1 ... an} , then Col(A)= Span {a1, ..., an}

Why is the PAP^-1 decomposition not unique?

The decomposition is not unique for several reasons: (1) the order of the eigenvalues λ=2,1 in D and the corresponding eigenvectors v1,v2 in P could be reversed; (2) the eigenvectors v1,v2 in P are not unique since scalar multiples of v1,v2 are again eigenvectors.

A set B = {v1, v2, . . . , vp} of vectors in R^n is always linearly dependent when p > n. True or Fase?

True

An n×n matrix A is diagonalizable if A has n distinct eigenvalues. True or False?

True

An n×n matrix is diagonalizable if and only if geo multA(λ) = alg multA(λ) for each eigenvalue λ of A. True or False?

True

If eigenvectors of an nxn matrix A are a basis for R^n, then A is diagonizable. True or False?

True

The only three-dimensional subspace of R^3 is R^3 itself. True or False?

True

For each y in R^n and each subspace W of R^n, the vector y − projection of y on W is in W⊥. True or False?

True I dont reall get this one, its on 1 of practice test

An n×n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. True or False?

True!

An n×n matrix A is invertible if and only if detA≠0. True or False

True!

The dimension of Nul A is the number of free variables in the equation Ax = 0. True or False?

True! It would be false to say: The dimension of Nul A is the number of variables in the equation Ax = 0.

If H is a p-dimensional subspace of R^n, then a linearly independent set of p vectors in H is a basis for H. True or False?

True; Because H is a p-dimensional subspace of R^n, any linearly independent set of exactly p elements in H is automatically a basis for H, hence also spans H.

If a set of p vectors spans a p-dimensional subspace H of R^n, then these vectors form a basis for H. True or False?

True; Because H is a p-dimensional subspace of R^n, any set of p elements of H that spans H is automatically linearly independent. Also, any linearly independent set of exactly p elements in H is automatically a basis for H.

If A is a 4 × 5 matrix, then dim(Col(A)) + dim(Nul(A)) = 5 . True or False?

True; By Fundamental Theorem of Linear Algebra, for an m × n matrix A, dim(Col(A)) + dim(Nul(A)) = n.

standard basis vectors are linearly independent?

Yes, because the matrix A= [e1,e2,e3,...en] is really In where In x = 0 has only trival solution with vector x=0 and matrix In is the identity matrix. And the Standard vectors for a basis of R^n and they are linearly independent

Find a basis for the eigenspace of the matrix A corresponding to given λ.

a basis for the eigenspace of the matrix A is the Nul(A-λI) of all solutions (A-λI)x=0

ignor

a dimension of 1

A set of vectors is said to be linearly dependent when

a nontrival solutions exists

consistant matrix

a system which has at least one solution is said to be consistant

Because two vectors v1,v2 in R^n are linearly dependent if and only if v2 is a multiple of v1, diagonalization is usually easy to check for 2×2-matrices.

after finding λ and plugging it back into the Ax=λx equation to achieve v1 and v2.

a pivot positions is

after row reduction, the position in a matrix contains a leading one

Diagonalization:

an n×n matrix A is said to be DIAGONALIZABLE when there exist a diagonal matrix and an invertible matrix P such that A=PDP^−1. When the linearly independent eigenvectors {v1,v1,...,vn} of A correspond to eigenvalues λ1,λ2,...,λn (not necessarily distinct), then A=PDP^−1 with P = [v1 v2 ⋯ vn] and pic

A set B={v1,v2,...,vn} of vectors is said to be an EIGENBASIS for R^n when there is ...?

an n×n matrix A such that B is a set of n linearly independent eigenvectors of A.

If B is an n×n matrix, then the homogeneous equation Bx=0 has non-trivial solutions if and only if...?

detB=0.

for a square matrix B the homogeneous equation Bx=0 has a non-trivial solution if and only if...?

detB=0. Thus a scalar λ is an eigenvalue of A if and only if λ is a solution of the scalar equation det(A−λI)=0, an eigenvector for A corresponding to λ is any non-trivial solution of the homogeneous matrix equation (A−λI)x=0.

det[AB]=...? Where A and B are arbitary nxn matrices

det[A]det[B].

det[A^−1]=...? when A is an invertible n×n matrix

det[A^−1]=(det[A])−1=1det[A]

Fundamental Theorem of Linear Algebra: Part I.

for an m×n matrix A: dim(Col(A))+dim(Nul(A)) = n.

inconsistant matrix

if it has no solutions

non-trivial solutions exists means...

implies that there are infinitely many solutions, so when a column has no pivot, the system has one free variable (and is also dependent system)

subspace of R^n is...

is a vector space that is a subset of some other (higher-dimension) vector space.

row reduction is ...?

is the process of using row operations to reduce a matrix to row reduced echelon form.

The null space of an m x n matrix A, written as Nul A, is the...?

set of all solutions to the homogeneous equation Ax = 0, using free variable terms s and t if needed to then find a solutions of the form Null(A)=Span{v1,v2}

the Algebraic Multiplicity of λ is ...?

the Algebraic Multiplicity of λ, denoted by alg multA(λ), is the number of times λ is repeated as a root of det(A−λI)=0, factor det(A−λI), then the Algebraic Multiplicity of eigenvalue α is number of times (λ−α) is repeated,

the geometric multiplicity of λ is ...?

the Geometric Multiplicity of λ, denoted by geo multA(λ), is the dimension of Nul(A−λI). compute rref(A−λI), then the Geometric Multiplicity is the number of free variables.

Span

the span of vectors v1, v2, ... , vn is the set of linear combinations: c1v1 + c2v2 + ... + cnvn and that this is in vector space.

A set of vectors: {v1, v2, v3, ... vn} is said to be linearly independent when ...

the vector equation: x1v1 + x2v2+...xpvp = 0 (where the x's are constants and the v's are vectors) has only trivial solutions. so matrix equation Ax has only trivial solutions.

trivial solutions exists means...

the zero vector is a solution

the solution of the initial value problem is...?

u(t)=(c1)(e^(λ1*t)(v1)+(c2)(e^(λ2*t)(v2)

Basis Property:

when a set B={v1,v2,...,vp} of vectors is a BASIS for a subspace H of R^n, then each x in H has a UNIQUE representation: x = c1v1+c2v2+...+cpvp in terms of the basis vectors.


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