Math 415 Exam 1

Existence and Uniqueness Theorem

a linear system is *consistent* (has a solution) if and only if an echelon form of the augmented matrix has NO ROW OF THE FORM [0.....0| b] where b is nonzero

an elementary row operation is :

a replacement (add multiple of one row to another) interchange(swap two rows) scaling (multiply a row by a scaler)

pivot variable

a variable that corresponds to a pivot column in the coefficient matrix of a system

free variable

a variable that is not a pivot variable

Uniqueness of the Reduced Echelon Form Theorem

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

the augmented matrix: [a1.....an | b] represents a consistent linear system if and only if b is expressible as a linear combination of the vectors a1.....an

true

the following expression is a linear combination in R2 -ln5 [-1 ,, 2] + e^3 [pi,,1] - [sin(pi/50 ,, sqrt2]

true

suppose, A,b,c are mxn matrices and r,s are in R. Which of the following are true...

A+B = B+A rA+rB = r(A+B) r(sA)=(rs)A (A+B)+C = A+(B+C)

[0,,,,,,,,,0] is a vector in every vector space

FALSE

let a,b,c,d be in R and A = [a,b,,c,d] then... A^3 = [a^3,b^3 ,, c^3, d^3]

FALSE

if A is a mxn matrix and B is a nxm matrix then AB is a nxn matrix

FALSE it would be a mxm matrix

every invertible matrix has an LU-decomposition

FALSE: [0,1,,1,0] is invertible but does not have an LU-decomposition

every square matrix has a LU decomposition

FALSE: the matrix [o,1,,1,0] does not

any line of plane in R^3 is a subspace of R^3

False: if the line or plane does not contain the origin, then it cannot be a subspace of R^3

every elementary matrix is invertible

TRUE

executing a row operation of matrix A is equivalent to multiplying A on the left by the elementary matrix E corresponding to the operation

TRUE

suppose A has a LU-decomposition and we want to solve Ax=b for many different bi, (say i = 1 thru 10^4). Then, first finding the LU-decomposition of A may be computationally beneficial

TRUE

let a,b,c,d be in R and A = [a,b,,c,d] then A is invertible if and only if ad-bc =/= 0

TRUE!

every triangular matrix, either upper or lower, has a LU-decomposition

TRUE! let I be the identity matrix of appropriate size. if A is lower triangular, then A=AI and if A is upper triangular then A=IA

if a linear system has more equations than variables, then the system must be inconsistent

false--> consider the possibility of a duplicated equation. in general consistency cannot be determined by just counting equations and variables, thus the benefit of echelon form

if the coefficient matrix of an augmented matrix has free variables, then the associated linear system has infinitely-many solutions

false--> the system may not be consistent so that talking about uniqueness is not.

the span of two non-zero vectors in R3 is geometrically a plane in R3

false: consider the possibility of parallel vectors

multiplying all entries in a row by a constant is an example of an elementary row operation

false: not true if the constant multiplier is zero

given a linear system, the number of solutions may be:

infinite, exactly one, or none

how many solutions will you get with any number of free variables

infinitely many solutions

which of the following CAN geometrically represent the span of a collection of vectors in R3

line, origin, plane all of R3

a linear system only has 3 options

one unique solution, no solution, or infinitely many solutions

pivot position

position of a leading entry in an echelon form of a matrix

the span of a set of vectors in R3 is a

subset of R3

pivot column

the column that contains a pivot position

Let P(n) be the vector space of polynomials of degree at most n. The subset of polynomials having 0 as a root is a subspace of P(n)

true

a linear system with m equations in n variables has a coefficient matrix with m rows and n columns

true

a nxn matrix A is invertible if and only if it is row-equivalent to the identity matrix I(n)

true

executing an elementary row operation on linear system leaves the set of solutions unchanged

true

if A,B are 3x3 matrices and x is the second column of B, then the second column of AB is Ax

true

if A,B are nxn matrices such that AB=I(n) then B=A^-1 and BA=I(n) also, where I(n) is the nxn identity matrix

true

let A be a mxn matrix and x,y are in R^n. if z is in span{x,y} then Az is in Span{Ax,Ay}

true

suppose A is a mxn matrix and x = [x1,,,,,,,,,,xn] in R^n. Then Ax is a linear combination of the columns of A with the corresponding weights x1......xn

true

the following is a linear system in the variables x(1) , x(2): pi*x(1) + (sin1)x(2) = -1 e^2 *x(1) + (ln(pi))x(2) = cos3

true

suppose A is an nxn matrix with LU-decomposition A=LU. if c is in R^n satisfies Lc=b and Uy=c, then y solves the equation Ax=b

true!

the product of lower triangular matrices is lower triangular

true!

the set of mxn matrices with real entries under matrix addition and scalar multiplication is an example of a vector space

true!

suppose L is lower-triangular. the system Ly=b may be solved by forward substitution

true! for lower-triangular coefficient matrix, the linear system is solved by forward substitution

if the coefficient matrix of an augmented matrix has pivots in every row, then the associated linear system must be consistent

true--> review the existence portion of the Existence and Uniqueness Theorem in the lecture on echelon forms

let A be an nxn matrix and x is in R^n. If A is not invertible, then the equation Ax=b is either inconsistent or has non-unique solutions

true: recall a matrix is invertible if and only if it is row-equivalent to an identity matrix

the product of permutation matrices is again a permutation matrix

true: the effect of doing successive permutations is again a permutation

suppose A is a mxn matrix and x is in R^n with m=/=n. then AX is a ...

vector in R^m review definition of matrix-vector multiplication

suppose A,B are invertible nxn matrices. Which of the following are also invertible?

AB A^-1 A^t NOT A+B

suppose r is in R and v is in R3. if A is a 5x3 matrix and B is a 3x5 matrix, which of the following expressions is defined?

B(Av) rA rv NOT: vB A(Bv) A+B

let A,B be nxn matrices and r is in R. which of the following are true?

(A+B)^t = A^t + B^t (rA)^t = rA^t NOT (AB)^t = A^t*B^t

let A,B be nxn matrices and r is in R. Which of the following are true?

(rAB) = (rA)B (AB)C = A(BC) (A+B)C = AC+BC NOT: AB = BA

suppose the coefficient matrix of a linear system has 5 rows and 7 columns. if there are 3 pivots, the sum of the number of pivots and free variables is

7 --> free variables are indexed by the columns not containing pivots

row echelon form

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.

if a linear system is consistent then the solution contains either...

1. a unique solution (where there are no free variables) or 2. infinitely many solutions (where there is at least 1 free variable)

ECHELON FORM (EF)

1. all rows full of zeros should be at the bottom 2. the leading entry of a nonzero row is always STRICTly to the right of the leading coefficient of the row below it 3. all entries in a column below a leading entry are zero

REDUCED ROW ECHELON FORM (RREF)

all the above and also 4. the leading entry in each nonzero row is 1 5. each leading 1 is the only nonzero entry in its column

R^n is a subspace of R^(n+1)

false