UMD Math 240 Final Exam

Shear

Transformation that stretches / compresses the shape of an object on one if its axes while keeping the others unchanged. Eg.

[1;k;0;1]

How to determine if vector b is in the range of transformation T

1. rref

2. if the system is consistent, it is, otherwise, it's not

How to use a single solution and the trivial solution to find the general solution?

Add the specific solution and the trivial solution

How to find the area/volume of a square matrix?

find the absolute value of the determinant

If V is a vector space of dimension n and S is a linearly independent set of n vectors in V, then S is a basis for V. (Always\Sometimes\Never)

Always

How to tell if a matrix is one-to-one?

Find the null space and check whether it contains only the zero vector.

How to tell if a transformation is onto?

1. RREF on the transformation matrix

2. check for linear independence

How to check if a matrix is linearly independent?

Find the determinant. If it's nonzero it is.

What is a bijective transformation?

The transformation is both one-to-one and onto.

Does surjective mean onto or one-to-one?

onto.

Does injective mean onto or one-to-one?

one-to-one.

How to create Q in QR factorization?

Do Gram-Schmidt and make it orthonormal

How to create R in QR factorization?

R = (Q^T)A

What does the Gram-Schmidt process do?

Generates an orthogonal set basis for any set of vectors.

Best approximation theorem

||y-yhat|| < ||y-v|| (v is any other point not yhat)

What is a singular matrix?

A matrix that is not invertible.

Two matrices are row equivalent if ...

they can be transformed into eachother through a sequence of elementary row operations.

If A is invertible, so is A^T (A transpose). (T/F)

True

If A is invertible, the columns of a span R^n.

True

The system Ax=0 has only the trivial solution if A (is/isn't) invertible.

Is.

How many solutions does the system Ax=b have for each b in R^n.

At least one solution for each b in R^n.

How many pivot positions must an nxn matrix have?

n

Best way to tell if a matrix is invertible.

If it's an nxn matrix and has n pivots.

If A is a triangular matrix, then det A is ...

the product of the left diagonal.

The determinant changes when adding a multiple of one row to another. (T/f)

False

If a row is swapped, the determinant of the matrix is __________.

multiplied by negative one.

If a row is multiplied by factor k, the determinant is multiplied by 1/k. (T/F)

False. It's just multiplied by k.

If A and B are nxn matrices, then det(AB) = _____.

det(A)*det(B)

Cramer's Rule

For an invertible, nxn matrix A, the solution to the system Ax=b can be given by x_i=(det(A_i(b))/(det(A)), i = i,...,n. Where A_i(b) means the column i is replaced with b.

Matrix Cofactor Formula

C_(ij) = (-1)^(i+j)*M_(ij)

Adjugate matrix definition

Transpose of cofactor matrix

Inverse of A formula with adjugate

A^(-1) = (1/(det(A)) * adjugate(A)

How to find a basis set?

Find the vectors that span the column space.

Conditions for a subspace

- Closed under addition >> if U and V are in the subspace, u+v must also be.

- Closed under scalar multiplication >> if u is in the subspace, c*u must also be.

- Contains zero vector

Is the vector space R^2 a subspace of R^3?

No. They are not of the same dimension so it cannot be a subspace. It also cannot contain the three dimensional zero vector.

Is a plane in R^3 that does not go thru the origin a subspace of R^3?

No, because it does not contain the zero vector.

The span of vectors v_1,...,v_n that form a vector space V, is not a subspace of V. (T/F)

False.

A vector being in the span of a set of other vectors means ...

the vector is a linear combination of the set of other vectors.

How to find col space

- RREF

- Col space is set of cols in the original matrix that correspond the pivot columns in the RREF matrix

How to find row space

- RREF

- Row space is set of rows in the original matrix that correspond to the nonzero rows in the RREF matrix

What is matrix rank?

Number of linearly independent rows/columns.

Is the number of linearly independent rows and linearly independent columns the same?

Yes. This is known as rank nullity theorem.

How to find null space?

- Augment to zero vector

- RREF

- Express in terms of free variables

Two properties of a linear transformation.

- T(a + b) = T(a) + T(b)

- T(c a) = c T(a)

If the columns of matrix A are linearly independent and span R^n then ...

they form a basis set for R^n.

How to show a system of equations in linearly independent (third way)

Equate it to the zero vector and show that it has only the trivial solution.

If a system has only the trivial solution, the system is linearly (independent / linearly dependent).

independent.

The pivot columns of a matrix A form a basis for _____.

Col(A)

If B = {b_1,...,b_n} is a basis set for vector space V, for each x in V, there exists a unique set of scalars:

x = (c_1)(b_1) + ... + (c_2)(b_2)

The coordinate vector x relative to b is :

[x]_B = [c_1(b_1); c_2(b_2) ; ... c_n(b_n)]

P_B * [x]_B =

x

(P_B)^-1 * x =

[x]_B

If B is a basis set for vector space V, the transformation from x to [x]_B is

one-to-one and linear.

What is an isomorphism?

A one-to-one, linear transformation from one vector space to another.

What is the dimension of the zero vector space?

Zero.

Dimension is the number of

linearly independent vectors (the size of the basis set).

If V is a p-dimensional vector space where is p ≥ 1, any linearly independent set of p elements that spans V is automatically,

a basis for V.

What are the two criteria for a basis set?

- Must span the vector space

- Must be linearly independent

The dimension of the null space is known as _______.

nullity

Rank(A) + Nullity(A) =

number of columns of A.

If A is invertible and the columns of A form a basis of R^n then the column space of A is

R^n.

If A is invertible and the columns of A form a basis of R^n then the rank of A is

n.

Similar matrices (do/ do not) have the same eigenvectors.

do not.

P_(B<-C) * [x]_B =

[x]_C

P_(C<-B) =

[ [b_1]_C ... [b_n]_C ]

(P_(C<-B))^-1 =

P_(B<-C)

[ c_1 c_2 | b_1 b_2 ] =

[ identity matrix | P_(C<-B) ]

The eigenvalues of a triangular matrix are the entries _____

of the main diagonal.

Eigenvectors of a given matrix are always linearly (dependent / independent)

independent.

Characteristic equation

(A - (lambda)(Identity matrix) * x = 0

Null space is the set of vectors v such that

T(v) = 0

Which property is preserved by linear transformation?

T(0) = 0

What do similar matrices have in common?

Eigenvalues, with the same multiplicities

How many solutions does least squares yield.

Either one or infinitely many.

If A is an m x n orthonormal matrix then (A^T) * A = I_n. (T/F)

True

D = [ a 0 ; 0 b], D^k = ?

[a^k 0 ; 0 b^k]

If nxn matrix A has n linearly independent eigenvectors, A (is / isn't) linearly independent.

is.

A = PDP^-1, what are D and P

- D is the a matrix of zeroes with the eigenvalues as the diagonals.

- P is the column matrix of eigenvectors.

If A is similar to B, A^k is similar to B^k. (T/F)

True

(A^-1) * (B^-1) =

(BA)^-1

How to check if matrices are similar?

P exists such that B = PAP^-1

Can an eigenvector of a linear transformation be the zero vector?

No.

[T(x)]_B =

M[x]_B

If B is the basis for R^n and we know A = PDP^-1, then D is the B-matrix for the transformation of

x onto Ax

If A is a real 2x2 matrix with complex eigenvalue lambda = a-bi, (b is not 0) and the associated eigenvector v, A = PCP^-1. What are P and C

P = [Re(v) Im(v)]

C = [a -b; b a]

The cross product is also known as ....

The dot product.

If and only if the dot product is zero, two vectors are ...

orthogonal.

|| u + v || ^2 = ||u||^2 + ||v||^2 means that u and v (are \ aren't) orthogonal.

Are.

A vector is in W_perp if and only if x is orthogonal to ...

every vector in the set that spans W.

(Row(A))_perp =

Nul(A)

(Col(A))_perp =

Nul(A^T)

angle between two vector as dot product u dot v =

||U|| ||V|| cos(theta)

If S is an orthogonal set of nonzero vectors in R^n then S is (linearly independent / linearly dependent) and (is / is not) a basis for the subspace spanned by S.

independent / is

If V = {u_1, ... u_n} is an orthogonal basis for the subspace of R^n, for each vector u in V the weights of the linear combination y = c_1(u_1) + c_2(u_2) + ... + c_n(u_n) can be given by c_i =

(y . u_i)/(u_i . u_i )

y-hat means

the orthogonal projection of y onto u

in the formula y = y_hat + z, z means

the component of y orthogonal to u

the distance from y to space L is given by

norm of z

An nxn matrix has u orthonomal columns if an only if

(U^T)U = I

||Ux|| =

||x||

U_x . U_y =

x . y