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