linear algebra exam FINAL

detA =____ when A is invertible

(-1)^r, product of pivots in U

λ is an eigenvalue of an n x n matrix A if and only if the equation ... has a nontrivial solution (a free variable).

(A - λI)x = 0

Theorem: Let A be an m x n matrix. The orthogonal complement of the row space of A is the null space of A, and the orthogonal complement of the column space of A is the null space of A^T:

(Row A) perp = NulA and (Col A) perp = Nul A^T

A scalar λ is an eigenvalue of an n x n matrix A if and only if λ satisfies the ....

characteristic equation, det(A - λI) = o

IF det(A - λI) is a polynomial in λ and if A is an n x n matrix, then det(A - λI) is a polynomial of degree n called the _________ of A

characteristic polynomial

Theorem: Let {u1, ..., up} be an orthogonal basis for a subspace W of R^n. For each y in W, the weights in the linear combination y = c1u1 + ... + cpup are given by cj =

cj = (y•uj)/(uj • uj) (j = 1, ... , p)

The _________ of an m x n matrix A, written as ColA, is the set of all linear combinations of the columns of A. If A = [a1 ... an] then ColA = Span{a1, ..., an}

column space

Suppose B = {b1, ..., bn} is a basis for V and x is in V. The ___________________

coordinates of x relative to the basis B (or the B-coordinated of x) are the weights c1, ..., cn such that x = c1b1 +... + cnbn.

Let A be an n x n matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling), and let r be the number of such row interchanges. Then the _____ of A, written as detA, is (-1)^r times the product of the diagonal entries u11, ..., unn in U. If A is invertible u11, ..., unn are all pivots (bc A ~ In)

determinant

A row replacement operation _______ change the determinant.

does not

The set of all solutions is the null space of the matrix A - λI. So this set is a subspace of R^n and is called the _____ of A corresponding to λ.

eigenspace

Theorem: An n x n matrix with n distinct _____ is diagonalizable.

eigenvalues

Theorem: If n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same _____ (with the same multiplicities)

eigenvalues

While row reduction was used to find eigenvectors, it cannot be used to find _____. An echelon form of a matrix A usually does not display the _____ of A.

eigenvalues

A is diagonalizable if and only if there are enough eigenvectors to form a basis of R^n. We call such basis an ____________ of R^n.

eigenvector basis

An _____ is one that is obtained by performing a single elementary row operation on an identity matrix.

elementary matrix

Each _____ E is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I.

elementary matrix

An indexed set of vectors {v1, ..., vp} in R^n is said to be _______ if the vector equation x1v1 + x2v2 + ... + xpvp = 0 has only the trivial solution.

linearly independent

An indexed set of vectors {v1, ..., vp} in V is said to be ________ if the vector equation has only the trivial solution, 0!

linearly independent

Theorem: The column space of an m x n matrix A is a ______ of R^m. Col A = {b: b = Ax for some x in R^n}

subspace

A _____ matrix is a matrix A such that A^T = A. Such. matrix is necessarily square. Its. main diagonal entries are arbitrary, but its other entries occurs in pairs - on opposite sides of the. main diagonal.

symmetric

If A is ______ then any two eigenvectors from different eigenspaces are orthogonal.

symmetric

Theorem: an n x n matrix A is orthogonally diagonalizable if and only if A is a ____ matrix.

symmetric

We say that two matrices are equal if...

they have the same size (same # of rows and columns) and if their corresponding columns are equal, which amounts to saying that their corresponding entries are equal.

A _______ (or function or mapping) T from R^n to R^m is a rule that assigns to each vector x in R^n a vector T(x) in R^m. The set R^n is called the ____ of T, and R^m is called the ____ of T.

transformation, domain, codomain

If A is a ____ matrix, then det A is the product of entries on the main diagonal of A.

triangular

Theorem: The eigenvalues of a _______ are the entries on its main diagonal.

triangular matrix

the zero solution is usually called the ______ solution

trivial

If y is in W = Span{u1, ..., up}, then projwy = y

..

detA = ____ when A is not invertible

0

dim Nul A =

0

Steps for diagonalizing matrices:

1. Find the eigenvalues of A. 2. Find linearly independent eigenvectors of A. 3. Construct P from the vectors in step 2. 4. Construct D from the corresponding eigenvalues.

If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form:

4. The leading entry in each nonzero row is 1. 5. Each leading 1 is the only nonzero entry in its column.

Theorem: The Best Approximation Theorem. Let W be a subspace of R^n, let y be any vector in R^n, and let ŷ be the orthogonal projection of y onto W. Then ŷ is the closest point in W to y, in the sense that || y - ŷ|| ___ ||y - v|| for all v in W distinct from ŷ.

<

[ T(x)]C =

= M[x]B

[x]B =

= P^-1x

P[x]B =

= x

For any scalar c, the length of cv is |c| times the length of v. That is, ||cv|| =

= |c| ||v||

For u and v in R^n, the distance between u and v, written as dist(u, v) is the length of the vector u -v. That is distinct(u, v) =

= ||u -v||

Theorem: The Pythagorean Theorem. Two vectors u and v are orthogonal if and only if || u + v ||^2 =

= ||u||^2 + ||v||^2

Theorem: Let A = [a b, c d]. If ad - bc does not equal 0, then A is invertible and ______. If ad - bc = 0, then A is not invertible.

A^-1 = 1/ad - bc [d -b, -c a].

Theorem: An inverse formula. Let A be an invertible n x n matrix. Then A^-1 =

A^-1 = 1/detA adjA

M = [ [T(b1)]C [T(b2)]C ... [T(bn)C]. The matrix M is a matrix representation of T, called the matrix for T relative to the bases ___ and ___.

B, C

When a Basis B is chosen for an n-dimensional vector space V, the associated coordinate mapping onto R^n provides a coordinate system for V. Each x in V is identified uniquely by its _________

B-coordinate vector [x]B.

Theorem: Diagonal Matrix Representation. Suppose A = PDP^-1, where D is a diagonal n x n matrix. If B I the basis for R^n formed from the columns of P, then D is the _____ for the transformation x --> Ax.

B-matrix

detA^T = (-1)detA

False. detA = detA^T

The _____ is a simple algorithm for producing an orthogonal or orthonormal basis for any nonzero subspace of R^n.

Gram-Schmidt Process

A^-1 A =

I, AA^-1 = I

Parallelogram Rule for Addition

If u and v in R^2 are represented as points in the plane, then u + v corresponds to the fourth vertex of the parallelogram whose other vertices are u, 0, and v.

If an elementary row operation is performed on an m x n matrix A, the resulting matrix can be written as EA, where the m x m matrix E is created by performing the same row operation on ___

Im

Theorem: Row Operations

Let A be a square matrix. a. If multiple of one row of A is added to another row to produce a matrix B, then detB = detA. b. If two rows of A are interchanged to produce B, then detB = -detA. c. If one row of A is multiplied by k to produce B, then detB = k x detA.

Theorem: Cramer's Rule

Let A be an invertible n x n matrix. For any b in R^n, the unique solution x of Ax = b has entries given by xi = detAi(b) / detA

Theorem: The Unique Representation Theorem

Let B = {b1, ..., bn} be a basis for a vector space V. Then for each x in V, there exists a unique set of scalars c1, ..., cn such that x = c1b1 + ... +cnbn

ColA =

R^n

The columns of A form a basis of ___

R^n

W perpendicular (perp) is a subspace of _____

R^n

Unlike the basis for Col A, the bases for ____ and ____ have no simple connection with the entries in A itself.

Row A and Nul A

A linear transformation T: R^n --> R^n is said to be invertible if there exists a function S: R^n --> R^n such that...

S(T(x)) = x for all x in R^n T(S(x)) = x for all x in R^n

______ is not the same as row equivalence. Row operations on a matrix usually change its eigenvalues.

Similarity. If A is row equivalent to B, then B = EA for some invertible matrix E.)

Pb^-1x = [x]b

Since the columns of Pb form a basis for R^n, Pb is invertible

Theorem: Let T: R^n --> R^m be a linear transformation. Then there exists a unique matrix A such that T(x) = Ax for all x in R^n. In fact, A is the m x n matrix whose jth column is the vector T(ej) where ej is the jth column of the identity matrix in R^n: A = [T(e1)... T(en)]

Theorem 10

The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A.

True, by definition.

An m x n matrix U has orthonormal columns if and only if

U^TU = I

[ c1 c2 | b1 b2] ~

[ I | P(c<--b)]

An efficient way to compute a B-matrix P^-1AP is to compute AP and then to row reduce the augmented matrix [ P | AP ] to ___________.

[ I | P^-1AP]

A vector equation x1a1 + x2a2 +...+ xnan = b has the same solution set as the linear system whose augmented matrix is ...

[ a1 a2 ... an b]

Theorem: Let T: R^n --> R^m be a linear transformation, and let A be the standard matrix for T. Then:

a. T maps R^n onto R^m if and only if the columns of A span R^m. b. T is one-to-one if and only if the columns of A are linearly independent.

Theorem: The determinant of any n x n matrix A can be computed by a _________ across any row or down any column. The expansion across the ith row using cofactors is detA = ai1Ci1 + ai2Ci2 + ... + ainCin. The cofactor expansion down the jth columns is detA = a1jC1j + a2jC2j + ... + anjCnj.

cofactor expansion

The ________ of an m x n matrix A is all of R^m if and only if the equation Ax = b has a solution for each b in R^m.

column space

A matrix with only one column is called a

column vector

Theorem: If a vector space V has a basis B = {b1, ..., bn}, then any set in V containing more than n vectors must be linearly _________

dependent

Suppose a square matrix A has been reduced to an echelon form U by row replacements and row interchanges. There are r interchanges. detA =

detA = (-1)^rdetU

Theorem: If A is an n x n matrix, then detA^T =

detA^T = detA

A row scaling scales the ______ by the same scalar factor.

determinant

A square matrix A is said to be _____ if A is similar to a diagonal matrix, that is, A = PDP^-1 for some invertible matrix P and some diagonal matrix D.

diagonalizable

Let H be a subspace of a finite-dimensional vector space V. Any linearly independent set in H can be expanded, if necessary to a basis for H. Also, H is finite-dimensional and dimH < or equal to

dimV

A row interchange ________ change the sign of the determinant.

does

An _____ of an n x n matrix A is a nonzero vector x such that Ax = λx for some scalar λ. A scalar λ is called an _____ of A if there is a nontrivial solution x of Ax= λx; such as x is called an eigenvector corresponding to λ.

eigenvector, eigenvalue

Theorem: The Diagonalization Theorem. An n x n matrix A is diagonalizable if and only if A has n linearly independent ______. In fact, A=PDP^-1, with D a diagonal matrix, if and only if the columns of P are n linearly independent eigenvectors of A. In this case, the diagonal entries of D are eigenvalues of A that correspond, respectively, to the eigenvectors in P.

eigenvectors

The dimension of Nul A is the number of ________ in the equation Ax = 0, and the dimension of Col A is the number of _______ in A.

free variables, pivot columns

a system of linear equations is said to be _____ if it has no solution

inconsistent

If u and v are vectors in R^n, then we regard u and v as n x 1 matrices. The transpose u^T is a 1 x n matrix, and the matrix product u^Tv is a 1 x 1 matrix. The number u^Tv is called the ________ of u and v, also mentioned as the dot product.

inner product

det A = (-1)^r (product of pivots in U) when A is ______.

invervtibe

number of pivot columns + number of non-pivot columns =

number of columns

Theorem: Let B = {b1, ..., bn} be a basis for a vector space V. Then the coordinate mapping x --> [x]b is a _________ linear transformation from V onto R^n.

one-to-one

A vector x is in W perpendicular if and only if x is _____ to every vector in a set that spans W

orthogonal

Two vectors u and v in R^n are _____ (to each other) if u • v = 0

orthogonal

If W is the subspace spanned by such an orthogonal set of unit vectors, then {u1, ..., up} is an ______ for W

orthogonal basis

The general least-squares problem is to find an x that makes ||b - Ax|| as _____ as possible

small

What are the 2 things that consists in an eigenspace?

the eigenspace consists of the zero vector and all the eigenvectors corresponding to λ.

if the augmented matrices of two linear systems are row equivalent, then ...

the two systems have the same solution set

Given an m x n matrix A, the ____ of A is the n x m matrix, denoted by A^T, whose columns are formed from the corresponding rows of A.

transpose

Theorem: If A is an invertible n x n matrix, then for each b in R^n, the equation Ax = b has the unique solution ...

x = A^-1b

The length (or norm) of v is the nonnegative scalar ||v|| defined by ||v|| =

||v|| = √v•v = √(v1)^2 +(v2)^2 + ... (vn)^2 and ||v||^2 = v•v

If two rows are interchanged, the determinant ______ signs

changes

Row interchange or swap ________ the sign of the determinant.

changes

det(A- λI) = 0 is the

characteristic equation

Let H be a subspace of a vector space V. An indexed set of vectors B = {b1, ..., bp} in V is a basis for H if:

(i) B is a linearly independent set, and (ii) the subspace spanned by B coincides with H; that is, H = Span{b1, ..., bp}.

A transformation (or mapping) T is linear if:

(i) T(u + v) = T(u) + T(v) for all u, v in the domain of T; (ii) T(cu) = cT(u) for all scalars c and all u in the domain of T.

Let V be a p-dimensional vector space, p> 1. What are the two things that are automatically a basis for V?

1) any linearly independent set of exactly p elements in V. 2) any set of exactly p elements that spans V

If λ + 5 is a factor of the characteristic polynomial of A, then 5 if an eigenvalue of A.

False, the eigenvalue is -5.

An elementary row operation on A does not change the determinant.

False. Interchanging rows or multiplying by a scalar changes the determinant.

If A is 3 x 3, with columns a1, a2, and a3, then detA equals the volume of the parallelepiped determined by a1, a2, and a3.

False. It is the absolute value of the determinant.

A row replacement operation on A does not change the eigenvalues.

False. They do change the eigenvalues.

P(c<--b) ^-1 =

P(b<--c)

No linear combination of vectors in R^3 can produce a vector in ...

R^4

(detA)(detB) = detAB

True

Theorem: Let B = {b1, ..., bn} and C = {c1, ..., cn} be bases of a vector space V. Then there is a unique n x n matrix P (C <--B) such that ... The columns of P(c<--b) are the C- coordinate vectors of the vectors in the basis b. That is, ....

[x]C = P(c<--b)[x]b. P(c<--b) = [ [b1]C [b2]C ... [bn]C]

Theorem: The Sprecral Theorem for Symmetric Matrices. An n x n symmetric matrix A has the following properties:

a. A has n real eigenvalues, counting multiplicities. b. The dimension of the eigenspace for each eigenvalue λ equals the multiplicity of λ as a root of the characteristic equation. c. The eigenspaces are mutually orthogonal, in the sense that eigenvectors corresponding to different eigenvalues are orthogonal. d. A is orthogonally diagonalizable.

Theorem: Properties of Determinants. Let A and B be n x n matrices.

a. A is invertible if and only if detA does not equal 0. b. detAB = (detA)(detB) c. detA^T = detA d. If A is triangular, then detA is the product of the entries on the main diagonal of A. e. A row replacement operation on A does not change the determinant. A row interchange changes the sign of the determinant. A row scaling also scales the determinant by the same scalar factor.

Theorem: The Spanning Set Theorem. Let S = {v1, ..., vp} be a set in V, and let H = Span{v1, ..., vp}.

a. If one of the vectors in S --vk-- is a linear combination of the remaining vectors in S, then the set formed from S by removing vk still spans H. b. If H does not equal {0}, some subset of S is a basis for H.

Theorem: Let A be an m x n matrix. The following statements are logically equivalent: When these statements are true, the least-squares solution x̂ is given by x̂ = (A^TA)^-1A^Tb

a. The equation Ax = b has a unique least-squares solution for each b in R^m. b. the columns of A are linearly independent c. The matrix A^TA is invertible.

Theorem: Let A be an n x n matrix whose distinct eigenvalues are λ1, ..., λp. a. For 1 <k <p, the dimension of the eigenspace for λk is less than or equal to the _____ of the eigenvalue λk. b. The matrix A is diagonalizable if and only if the sum of the dimensions of the ______equals n, and this happens if and only if (i) the characteristic polynomial factors completely into linear factors and (ii) the dimension of the eigenspace for each λk equals the multiplicity of λk. c. If A is diagonalizable and Bk is a basis for the eigenspace corresponding to λk for each k, the the total collection of vectors in the sets B1, ..., Bp forms an _______ basis for R^n.

a. multiplicity b. eigenspaces c. eigenvector

Theorem: Let, u, v, and w be vectors in R^n, and let c be a scalar. Then...

a. u • v = v • u b. (u + v)• w = u•w + v•w c. (cu)•v = c(u•v) = u•(cv) d. u•u > 0, and u•u = 0 if and only if u = 0.

Let U be an m x n matrix with orthonormal columns, and let x and y be in R^n. Then...

a. ||Ux|| = ||x|| b. (Ux) ⋅ (Uy) = x ⋅ y c. (Ux) ⋅ (Uy) = 0 if and only if x ⋅ y = 0

Theorem: If S = {u1, ..., up} is an orthogonal set of nonzero vectors in R^n, then S is linearly independent and hence is a _____ for the subspace spanned by S.

basis

Theorem: Let V be a p-dimenstional vector space, p > 1. Any linearly independent set of exactly p elements in V is automatically a ____ for V. Any set of exactly p elements that spans V is automatically a ____ for V.

basis

Theorem: The pivot columns of a matrix A form a _____ for ColA

basis

a ______ of V is a linearly independent set that spans V.

basis

the dimension of V is the number of vectors in a _____ for V

basis

x = Pb[x]b

change - of - coordinates matrix

Pb = [ b1 b2 ... bn] where Pb is the

change-of-coordinates matrix

The matrix P(c<--b) is called the _________. Multiplication by P(c<--b) converts B-coordinates into C-coordinates.

change-of-coordinates matrix from B to C.

If V is spanned by a finite set, then V is said to be ____________, and the dimension of V, written as dim V, is the number of vectors in a basis for V. The dimension of the zero vector space {0} is defined to be zero. If V is not spanned by a finite set, then V is said to be __________

finite-dimensional, infinite-dimensional

Theorem: If v1, ..., vr are eigenvectors that correspond to distinct eigenvalues λ1, ..., λr of an n x n matrix A, then the set {v1, ..., vr} is linearly ________

independent

One row is multiplied by a scalar. The determinant _____

is also multiplied by a scalar.

In general, a one-to-one linear transformation from a vector space V onto a vector space W is called an _____ from V onto W.

isomorphism

The ______ (or null space) of such a T is the set of all u in V such that T(u) = 0 (the zero vector in W).

kernel

If A is m x n and b is in R^m, a _____ of Ax = b is an x̂ in R^n such that || b - Ax̂ || < || b - Ax|| for. all x in R^n.

least-squares solution

A ___________ T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W, such that (i) T(u + v) = T(u) + T(v) for all u, v in V, and (ii) T(cu) = cT(u) for all u in V and all scalars c.

linear transformation

The set {v1, ..., vp} is said to be _______ if the vector equation has a nontrivial solution, that is, if there are some weights c1,..., cp, not all zero, such that the vector equation holds.

linearly dependent

Theorem: An indexed set {v1, ..., vp} of two or more vectors, with v1 not equalling zero, is _______ if and only if some vj (with j > 1) is a linear combination of the preceding vectors.

linearly dependent

Theorem: The Invertible Matrix Theorem (cont.) Let A be an n x n matrix. Then the following statements are each equivalent to the statement that A is an invertible matrix.

m. The columns of A form a basis of R^n. n. ColA = R^n o. dimColA = n p. rankA = n q. Nul A = {0} r. dimNulA = 0

In general , the ______ of an eigenvalue λ is its _____ as a root of the characteristic equation.

multiplicity

Theorem: If a vector space V has a basis of n vectors, then every basis of V must consist of exactly __ vectors

n

dimColA =

n

rank A =

n

An ______ for a subspace W of R^n is a basis for W that is also an orthogonal set.

orthogonal basis

The set of all vectors z that are orthogonal to W is called to the __________ of W and is denoted by W perpendicular

orthogonal complement

An ___________ is a square invertible matrix U such that U^-1 = U^T

orthogonal matrix

the vector ŷ in y = ŷ + z is called the ___________ and is written as projwy.

orthogonal projection of y onto W

The vector ŷ is called the ___________

orthogonal projection of y onto u

A set of vectors {u1, ..., up} in R^n is said to be an ________ if each pair of distinct vectors from the set is orthogonal, that is, if ui • uj = 0 whenever i does not equal j

orthogonal set

If a vector z is orthogonal to every vector in a subspace W of R^n, then z is said to be _______ W

orthogonal to

An n x n matrix A is said to be ___________ if there is an orthogonal matrix P (with P^-1 = P^T) and a diagonal matrix D such that A= PDP^T = PDP^-1. Such a diagonalization requires n linearly independent and orthonormal eigenvectors.

orthogonally diagonalizable

The ______ of T is the set of all vectors in W of the form T(x) for some x in vector V.

range

The ____ of A is the dimension of the column space of A.

rank

Theorem: The Rank Theorem. The dimensions of the column space and the row space of an m x n matrix A are equal. This common dimension, the rank of A, also equals the number of pivot positions in A and satisfies the equation...

rank A + dimNulA = n

The set of all linear combinations fo the row vectors is called the ______ of A and is denoted by Row A. RowA is a subspace of R^n because each row has n entries.

row space

Theorem: Invertible Matrix Theorem (continued... again). Let A be an n x n matrix. Then A is invertible if and only if

s. The number 0 is not an eigenvalue of A. t. The determinant of A is not zero.

If 2 matrices A and B are row equivalent, then their row spaces are the ____. If B is in echelon form, the nonzero rows of B form a basis for the row space of A as well as for that of B.

same

A vector whose length is 1 is called a ______

unit vector

A set {u1, ..., up} is an orthogonal set if it is an orthogonal set of ______

unit vectors

Theorem: The orthogonal decomposition theorem. Let W be a subspace of R^n. Then each y in R^n can be written uniquely in the form _____ where ŷ is in W and z is in W perp. IN fact, if {u1, ..., up} is any orthogonal basis of W, then ŷ = (y * u1)/(u1 * u1) u1 + ... _ (y*up)/(up*up) up and z = y - ŷ

y = ŷ + z

Given a vector y and a subspace W in R^n, there is a vector ŷ in W such that (1) ŷ is the unique vector in W for with y - ŷ is orthogonal to W, and (2) ŷ is the unique vector in W closest to y.

yes

y = ŷ + z

yes

ŷ = (y ⋅ u)/ (u ⋅ u) x u

yes

A^TAx = A^Tb

yes.

Theorem: If{u1, ..., up} is an orthonormal basis for a subspace W of R^n, then projwy = (y*u1)u1 + (y*u2)u2 + ... + (y*up)up. If U = [ u1 u2 ... up], then projwy = UU^T for all y in R^n.

yes.

When a least-squares solution x̂ is used to produce Ax̂ as an approximation to b, the distance from b to Ax̂ is called the least-squares error of this approximation.

yes.

Theorem: The set of least-squares solutions of Ax =. b coincides with the nonempty set of solutions of the normal equations A^TAx = A^Tb

yup.

Nul A =

{0}

sometimes ŷ is denoted by projLy and is called the orthogonal projection of y onto L. That is, ŷ =

ŷ = projL y = (y ⋅ u)/ (u ⋅ u) x u

A rectangular matrix is in echelon form (or row echelon form) if it has the 3 following properties:

1. All nonzero rows are above any rows of all zeros 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros.

Theorem: Existence and Uniqueness Theorem

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column-- that is, if and only if an echelon form of the augmented matrix has no row of the form [ 0 ... 0 b] with b nonzero. If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable.

If A is an m x n matrix, and if B is an n x p matrix with columns b1, ..., bp, then the product AB is the m x p matrix whose columns are Ab1, ..., Abp. That is... AB =

AB = A[b1 b2 ... bp] = [Ab1 Ab2 Abp]

Theorem: The Invertible Matrix Theorem

Let A be a square n x n matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false. a. A is an invertible matrix b. A is row equivalent to the n x n identity matrix. c. A has n pivot positions. d. The equation Ax = 0 has only the trivial solution. e. The columns of A form a linearly independent set. f. The linear transformation x --> Ax is one-to-one. g. The equation Ax = b has at least one solution for each b in R^n. h. The columns of A span R^n. i. The linear transformation x --> Ax maps R^n onto R^n. j. There is an n x n matrix C such that CA = 1. k. There is an n x n matrix D such that AD = I. l. A^T is an invertible matrix.

Theorem: a) If A is an invertible matrix, then A^-1 is inveritible and (A^-1)^-1 = A. b) If A and B are n x n invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. That is, (AB)^-1 = B^-1A^-1. c) If A is an invertible matrix, then so is A^T, and the inverse of A^T is the transpose of A^-1. That is, (A^T)^-1 = (A^-1)^T

Theorem 6

Theorem: Let T: R^n --> R^n be a linear transformation and let A be the standard matrix for T. Then T is invertible if and only if A is an invertible matrix. In that case, the linear transformation S given by S(x) = A^-1x is the unique function satisfying equations S(T(x)) = x for all x in R^n and T(S(x)) = x for all x in R^n.

Theorem 9

If A is an m x n matrix, with columns a1, ..., an, and if b is in R^m, the matrix equation Ax = b has the same solution set as the vector equation x1a1 + x2a2 + ... + xnan = b, which, in turn, has the same solution set as the system of linear equations whose augmented matrix is

[ a1 a2 ... an b]

a matrix that is not intervible is sometimes called a ______

a singular matrix

Theorem: Let A and B denote matrices whose sizes are appropriate for the following sums and products. ...

a) (A^T)^T b) (A + B)^T = A^T + B^T c) For any scalar r, (rA)^T = rA^T d) (AB)^T = B^TA^T

Theorem: Let A be an m x n matrix. Then the following statements are logically equivalent. That is, for a particular A either they are all true statements or they are all false. (Warning: this is about a coefficient matrix, not an augmented matrix. If an augmented matrix [A b] has a pivot position in every row, then the equation Ax = b may or may not be consistent.

a. For each b in R^m, the equation Ax = b has a solution. b. Each b in R^m is a linear combination of the columns of A. c. The columns of A span R^m. d. A has a pivot position in every row.

Let a1 and a2 be nonzero vectors. Then for any scalar c, the ____ of the parallelogram determined by a1 and a2 equals the area of the parallelogram determined by a1 and a2 + ca1.

area

The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has ...

at least one free variable.

The equation Ax = b has a solution if and only if

b is a linear combination of the columns of A.

If a product AB is the zero matrix, you _____ conclude in general that either A = 0 or b = 0

cannot

a system of linear equations is said to be _____ if it has either one solution or infinitely many solutions

consistent

Theorem: Multiplicative Property. If A and B are n x n matrices, then detAB =

detAB = (detA)(detB)

For n>2, the ____ of an n x n matrix A = [aij] is the sum of n terms of the form +/- a1j detA1j, with plus and minus signs alternating, where the entries a11, a12, ..., a1n are from the first row of A. In symbols, det A = a11detA11 - a12detA12 + ... + (-1)^1 + n a1ndetA1n

determinant

The quantity ad - bc is called the _____ of A. det A = ad - bc.

determinant

A ______ is a square n x n matrix whose non diagonal entries are zero. example is the n x n identity matrix, In.

diagonal matrix

One row times a coefficient added to another row, the determinant ______

does not change

In general, AB ____ BA

does not equal

Theorem: A square matrix A is invertible if and only if detA ...

does not equal 0.

Theorem: Uniqueness of the Reduced Echelon Form

each matrix is row equivalent to one and only one reduced echelon matrix

If a linear system is consistent, then the solution is unique if and only if...

every column of the coefficient matrix is a pivot column; otherwise, there are infinitely many solutions.

A system of linear equations is said to be _____ if it can be written in the form Ax = 0, where A is an m x n matrix and 0 is the zero vector in R^m. This system always has at least one solution, namely x = 0 (the zero vector).

homogeneous

Theorem: Supposed the equation Ax=b is consistent for some given b, and let p be a solution. Then the solutions set of Ax=b is the set of all vectors of the form w = p + vh, where vh is any solution of the... Warning: only applies to an equation Ax = b that has at least one nonzero solution p. When Ax= b has no solution, the solution set is empty.

homogeneous equation Ax = 0.

The only two row operations that change the determinant are ...

interchanging rows and multiplying by a scalar

An n x n matrix A is said to be _____ if there is an n x n matrix C such that CA = I and AC = I

invertible

Let A and B be square matrices. If AB = I, then A and B are both _____, with B = A^-1 and A = B^-1.

invertible

If A is an m x n matrix, with columns a1, ..., an, and if x is in R^n, then the product of A and x, denoted by Ax, ...

is the linear combination of the columns of A using the corresponding entries in x as weights = x1a1 + x2a2 + ... + xnan

_________ _______ refers to any sum of scalar multiples of vectors, and Span{v1, ..., vp} denotes the set of all vectors that can be written as _____ _______ of v1, ..., vp.

linear combinations

Every matrix transformation is a _______

linear transformation

A set of two vectors {v1, v2} is ________ if at least one of the vectors is a multiple of the other.

linearly dependent

The set of vectors {v1, ..., vp} is said to be _______ if there exists weights c1, ..., cp, not all zero, such that c1v1 + c2v2 + ... + cpvp = 0

linearly dependent

Theorem: If a set S = {v1, ..., vp} in R^n contains the zero vector, then the set is _______.

linearly dependent

Theorem: If a set contains more vectors than there are entries in each vector, then the set is _______. That is, any set {v1, ..., vp} in R^n is _______ if p > n.

linearly dependent

A set of two vectors {v1, v2} is ________ if and only if neither of the vectors is a multiple of the other.

linearly independent

Row operation is row replacement and it has _______ on the determinant.

no effect

an invertible matrix is called a ____

nonsingular matrix

_____ is a nonzero vector x that satisfies Ax= 0

nontrivial solution

detA = 0 when A is _________

not invertible

The _________ of an m x n matrix A, written as NulA, is the set of all solutions of the homogeneous equation Ax = 0. NulA = {x: x is in R^n and Ax = 0}

null space

A mapping T: R^n ---> R^m is said to be ___ if each b in R^m is the image of at most one x in R^n.

one-to-one

T is _____ if, for each b in R^m, the equation T(x) = b has either a unique solution or none at all. The mapping T is NOT ______ when some b in R^m is the image of more than one vector in R^n.

one-to-one

Theorem: Let T: R^n --> R^m be a linear transformation. Then T is ______ if and only if the equation T(x) = 0 has only the trivial solution.

one-to-one

A mapping T: R^n ---> R^m is said to be ___ R^m if each b in R^m is the image of at least one x in R^n.

onto

a _______ in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A ______ is a column of A that contains a pivot position.

pivot position, pivot column

The product of n x n invertible matrices is invertible, and the inverse is the ...

product of their inverses in the reverse order.

augmented matrix

put the constants on the right-hand side next to the coefficient matrix

elementary row operations

replacement (replace one row by itself and a multiple of another row), interchange (interchange 2 rows), scaling (multiply all entries in a row by a nonzero constant)

The transpose of a product of matrices equals the product of their transposes in the ____ order.

reverse

An n x n matrix A is invertible if and only if A is _____ , and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A^-1.

row equivalent

Algorithm for Finding A^-1

row reduce the augmented matrix [A I]. If A is row equivalent to I, then [A I] is row equivalent to [I A^-1]. Otherwise, A does not have an inverse.

If r is a scalar and A is a matrix, then the _____ rA is the matrix whose columns are r times the corresponding columns in A.

scalar multiple

The transformation T: R^2 to R^2 defined by T(x) = Ax is called a ______ transformation

shear

A = [T(e1) .... T(en)] is called the ______

standard matrix for the linear transformation T.

If v1, ..., vp are in R^n, then the set of all linear combinations of v1, ..., vp is denoted by Span{v1, ..., vp} and is called the

subset of R^n spanned (or generated) by v1, ..., vp. That is Span{v1, ..., vp} is the collection of all vectors that can be written in the form c1v1 + c2v2 + ... + cpvp with c1,..., cp scalars.

A _____ of a vector space V is a subset H of V that has three properties: a. The zero vector of V is in H. b. H is closed under vector addition. That is, for each u and v in H, the sum u + v is in H. c. H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector cu is in H.

subspace

The null space of an m x n matrix A is a ______ of R^n. Equivalently, the set of all solutions to a system Ax = 0 of m homogeneous linear equations is n unknowns is a subspace of R^n.

subspace

Theorem: If v1, ...., vp are in vector space V, then Span{v1, ..., vp} is a _____ of V.

subspace

The columns of a matrix A are linearly independent if and only if the equation Ax= 0 has only the _____

trivial solution

A ______ is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the ten axioms listed below. The axioms must hold for all vectors u, v, and w in V and for all scalars c and d. 1. The sum of u and v is in V. 2. u + v = v + u 3. (u + v) + w = u + (v + w) 4. There is a zero vector 0 in V such that u + 0 = u 5. For each u in V, there is a vector -u in V such that u + (-u) = 0 6. The scalar multiple of u by c, denoted by cu, is in V. 7. c(u + v) = cu + cv 8. (c + d)u = cu + du 9. c(du) = (cd)u 10. 1u = u

vector space

If A is an n x n matrix and E is an n x n elementary matrix, then deter = (detE)(detA) where detE = 1 if E is a row replacement, -1 if E is an interchange, and r if E is a scale by r.

yes.

An m x n matrix whose entries are all zero is a ____ and is written as 0.

zero matrix

the set consisting of only the zero vector in a vector space V is a subspace of V, called the ______ and written as {0}

zero subspace

The vector whose entries are all zero is called the

zero vector and is denoted by 0.

Theorem: Let T: R^2 --> R^2 be the linear transformation determined by a 2 x 2 matrix A. If S is a parallelogram in R^2, then, {area of T(S)} = ______________. If T is determined by a 3 x 3 matrix A, and if S is a parallelepiped in R^3, then {volume of T(S)} = ______________.

{area of T(S)} = | detA | x {area of S}. {volume of T(S)} = | detA | x {volume of S}.

Theorem: If A is a 2 x 2 matrix, the area of the parallelogram determined by the columns of A is _____. If A is a 3 x 3 matrix, the volume of the parallelepiped determined by the columns of A is _____.

|det A |