emth211 written test

The product of a column vector and a row vector resulting in a matrix

Outer Product

A partitioned matrix with non-zero blocks along the main diagonal and zero matrices elsewhere

Block Diagonal Matrix

A special matrix that performs an elementary row operation on a target matrix when multiplied on the left

Elementary Matrix

Gaussian Elimination

An algorithm to solve a system of linear equations in two steps being row reduction and back substitution

Row Echelon Form

A matrix state where all zero rows are at the bottom and each pivot is to the right of the row above

Pivot

The first non-zero entry in each row of a matrix in row echelon form

Upper Triangular Matrix

A square matrix in row echelon form where all elements below the main diagonal are zero

what are the Elementary Row Operations

The three safe row operations

Pivot Variable

A variable corresponding to a column containing a pivot whose value can be solved for

What is a free Variable

A variable corresponding to a column without a pivot whose value is arbitrary

Fixed Point Numbers

Numbers with fixed whole number precision and fixed fractional precision that have limited practical use

Floating Point Numbers

Numbers represented by a sign

A 32-bit floating point representation available in NumPy as np float32

Single precision

Double Precision

A 64-bit floating point representation which is Python and NumPy default

Round Off Error

Information loss caused in almost every operation due to limited precision arithmetic

Backward Error Analysis

An approach where the error in a computer computation is pushed back and placed on the data as a perturbed system

Small Pivots

Values that cause errors to be heavily magnified during division steps in Gaussian elimination

Partial Pivoting

A strategy where rows are swapped so the pivot always has the largest absolute value in that column

Complete Pivoting

A strategy using both row and column swaps to place the largest absolute value in the submatrix into the pivot position

Ill-conditioned System

A system where small data inaccuracies lead to massive deviations in the computed solution

Flops

Floating point operations used to measure the computational speed and cost of an algorithm

Flop Count for Row Reduction

The total number of operations required during row reduction scaling at O 2 thirds n cubed

Flop Count for Back Substitution

The number of operations required during back substitution scaling at O n squared

Row Vector

A matrix with dimensions 1 by n

Column Vector dimensions

m by 1

what does Transpose mean

An operation that flips a matrix over its diagonal turning rows into columns

Outer Product

The product of a column vector and a row vector resulting in a matrix

Block Diagonal Matrix

A partitioned matrix with non-zero blocks along the main diagonal and zero matrices elsewhere

Elementary Matrix

A special matrix that performs an elementary row operation on a target matrix when multiplied on the left

What is the span of a set of vectors?

The collection of all linear combinations of those vectors

What is the column space of matrix A?

The span of the columns of A

What is the null space of matrix A?

The set of all solutions to the homogeneous matrix equation Ax = 0

When does Ax = b have a solution?

When vector b belongs to the column space of A

What makes a set of vectors linearly independent?

The only solution to their linear combination equaling zero is if all coefficients are zero

What is a basis for a subspace?

A linearly independent spanning set

What is the dimension of a subspace?

The number of vectors in its basis

What is the nullity of a matrix?

The dimension of its null space

What is the rank theorem for an m by n matrix?

Rank(A) plus nullity(A) equals n

How do you find a basis for the row space?

Find the RREF of A and take the non-zero rows

How do you find a basis for the column space?

Find the RREF to identify pivot columns and use those original columns from A

What subspace is orthogonal to the null space?

The row space

What defines an orthogonal set of vectors?

The dot product of any two distinct vectors in the set is zero

What defines an orthonormal set?

An orthogonal set where every vector has a length of one

What is an orthogonal matrix?

A square matrix whose columns form an orthonormal set

What is the inverse of an orthogonal matrix Q?

Its transpose

What does an orthogonal transformation preserve?

Both the length of vectors and the angles between them

What is the purpose of the Gram-Schmidt process?

To convert an arbitrary basis into an orthogonal or orthonormal basis

How do you find the projection of x onto an orthonormal basis vector q?

Take the dot product of x and q and multiply by q

What is an idempotent matrix?

A matrix A where A squared equals A

What two properties define a projection matrix?

It is symmetric and idempotent

What is the projection matrix P for a unit vector q?

q times q transpose

What is the projection matrix P for a subspace with orthonormal basis Q?

Q times Q transpose

What is the null space of a projection matrix P orthogonal to?

The column space of P

What does QR factorization decompose matrix A into?

An orthogonal matrix Q and an upper triangular matrix R

Why must the diagonal elements of R in QR factorization be non-zero?

Because the columns of A are linearly independent

How do you solve Ax = b using QR factorization?

Solve Rx = Q transpose times b using back substitution

What is the orthogonal complement of a subspace W?

The set of all vectors orthogonal to every vector in W

What is the orthogonal complement of the column space of A?

The null space of A transpose

What is the orthogonal complement of the row space of A?

The null space of A

What does the Orthogonal Decomposition Theorem state?

Any vector v can be uniquely written as w plus w perpendicular

What happens if you project a vector in the column space of P using P?

The vector remains unchanged

What form is a linear system rewritten into for fixed point iteration?

x = g(x)

What matrix splitting is used for iterative linear solvers?

A = P − Q

What is the general iterative scheme?

Px^(k+1) = Qx^(k) + b

What makes Jacobi iteration different from Gauss-Seidel?

Jacobi uses only old values

What does Gauss-Seidel use when computing later components?

Newly computed values

What type of matrix is P in Jacobi iteration?

Diagonal

What type of matrix is P in Gauss-Seidel iteration?

Triangular

What condition is required on the main diagonal for Jacobi and Gauss-Seidel?

No diagonal entries can be zero

What decomposition splits A into lower diagonal and upper parts?

A = L + D + U

What is the Jacobi iteration matrix?

M = −D⁻¹(L + U)

What is the lower triangular Gauss-Seidel iteration matrix?

M = −(L + D)⁻¹U

What condition guarantees convergence of an iteration in some norm?

||M|| < 1

What happens to the error during fixed point iteration?

It is multiplied by M

What matrix property guarantees Jacobi convergence?

Strict diagonal dominance

What does strict diagonal dominance mean?

Diagonal magnitude exceeds other row magnitudes combined

Is diagonal dominance necessary for Jacobi convergence?

No it is only sufficient

Which method generally converges faster?

Gauss-Seidel

Can rearranging equations help iterative methods converge?

Yes it can create diagonal dominance

What is a sparse matrix?

A matrix with mostly zero entries

What is a tridiagonal matrix?

Nonzero entries only on three central diagonals

Why is Gaussian elimination preferred for tridiagonal systems?

It is very efficient

What is a vector space closed under?

Linear combinations

Why must a vector space contain the zero vector?

Scalar multiplication by zero produces it

Why is the positive quadrant not a vector space?

It is not closed under negative scaling

What is a subspace?

A vector space contained within another vector space

Can a subspace equal its parent vector space?

Yes

What does rank equal in REF or RREF?

The number of pivots

Does the choice of REF change matrix rank?

No

What extra conditions define RREF?

Pivots are one with zeros above and below

What does Ax = b having a solution mean geometrically?

b is in the column space of A

What is the column space?

All linear combinations of the columns of A

What does an underdetermined system have?

More columns than rows

Can an underdetermined system have a unique solution?

No

What does an overdetermined system have?

More rows than columns

What can an overdetermined system contain?

Redundant equations or no solution

For a consistent system what is the number of free variables?

n − rank(A)

What does each nonpivot column correspond to?

A free variable

What is a linear combination?

A sum of scalar multiples of vectors

How can linear combinations be found?

By solving a matrix equation