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Sample Problems Math 310
Gaussian Elimination and Row-Echelon Form
Use the Reduced Row-Echelon form to find all solutions of the equations Problem 1:
3 = z + y 3 + x
8 = z + y 5 + x 2
11 = z 2 + y 8 + x 3
You must show all your steps and work for credit.
Find the general solution of Problem 2:
(^1) x
x 2
(^3) x
(a) Find the row-reduced echelon form of Problem 3:
= A
= 0? (Check!) Ax (b) What are the solutions of the system
Given the equations Problem 4:
1 = w 3 − z + 3 y + 2 x
1 = w 6 − z + 6 y + 5 x 4
1 = w 8 − z + 9 y + 8 x 7
a) Give the Reduced Row-Echelon form of the associated augmented matrix.
b) Which are the free variables? Which are the dependent variables?
c) Give the general solution of the system of equations.
Given the two equations Problem 5:
2 = w 4 − z + 3 y + 2 x
5 = w + z + 3 y + 4 x 2
Use the method of row reduction to solve the system. Indicate which are the free variables, which are the
dependent variables. What is the geometric interpretation of the solution?
be constants, and consider the system of equations a, b, c Let Problem 6:
a = z + y 3 + x 3
b = z 2 + y + x
c = y 5 + x 5
must satisfy so that these equations are consistent. a, b, c Find the equation that the constants
Matrix Algebra and Manipulating Matrices
In each case, give an example of a matrix which is Problem 1:
not the identity matrix •
, not the zero matrix •
and satisfies:
2 diagonal matrix with an inverse. × is a 2 A a)
2 matrix with rank 1. × is a 2 B b)
2 symmetric matrix with no inverse. × is a 2 C c)
2 orthogonal matrix. × is a 2 O d)
.O and C , B ,A So, you must find four matrices
Matrix Determinants
= A Find the determinant of the matrix Problem 1:
Use either the definition of determinant in terms of cofactors, or the method of row operations, Problem 2:
to calculate the determinant of
= A
= B Calculate the determinant of the matrix Problem 3:
A Find the determinant of the matrix Problem 4: 3
= A where
[
]
= A Given the matrices Problem 5:
[
]
= B ,
[
]
= C ,
[
]
calculate the following determinants:
|C| and | B| ,|A| a)
ABC| b) 2
Matrix Inverses
= A a) Find the inverse (by any method) of Problem 1:
[
]
= xA~ b) Use the above to express the solutions of ~
.^2 b and^1 b in terms of the constants b
= A Give the formula for the inverse of Problem 2:
[
b a
d c
]
= A Use the method of Gaussian Elimination to find the inverse for Problem 3:
= A Use the method of Cofactors to find the inverse for Problem 4:
Find the inverse of the following matrices (and check your answers.) Problem 5:
Do not use a calculator – you will be required to show all your work and computations.
= C a)
= C b)
= A c)
= A d)
Explain your below have an inverse? D does the matrix λ For what values of the variable Problem 6:
answer!
= D
1 3 λ − 3
5 λ − 2 0
= 0 has a unique AX Suppose that the system of equations matrix. n × n be an A Let Problem 7:
A solution. Explain why the inverse 1 −
has to exist.
Vector Spaces and Subspaces
R Consider the subset of vectors in Problem 1: 2
given by
x, x({ = S 2
} ) where x is any real number
a vector subspace? Justify your answer carefully. S Is
Is the set Problem 2:
{[
x
x 3
]
R ∈ x where
R a vector subspace of 2
? Justify your answer.
of the equation S. Show the solution setx be the space of real-valued functions of V Let Problem 3:
f ′
)x( xf ) =x(
. V is a subspace of
be the subset of all W Let be the space of all differentiable functions on the line. V Let Problem 4:
f which are solutions of the differential equation f functions ′′
is W = 0. Show that the solution set f + 5
. V a subspace of
A Let Problem 5: n×m
What are the four fundamental columns. n rows and m be a matrix with
? Give the definition of each of the following:A subspaces associated to
. A ) = the column space ofA Col( •
A ) = the row space ofA Row( •
A ) = the null space ofA Null( •
A ) = conull space ofA Conull( •
Column Space, Row Space, Null Space, Conull Space
matrix. Let n × m is an A Problem 1:
A ) denote the column space ofA Col( •
A ) denote the row space ofA Row( •
A ) denote the null space ofA Null( •
A ) the co-null space ofA Conull( •
For each of the following questions, your answer should be one of the above 4 spaces. Justify your answer
by stating why you think it is correct.
a) The set of vectors perpendicular to the column space is what space?
= xA~ b) The vector equation ~
has a solution if b ~
belongs to what subspace? b
c) The set of vectors perpendicular to the row space is what space?
= xA~ d) The vector equation ~
?} 0 { has a unique solution if what space is b
e) What number do you get if you add the dimensions of all 4 spaces?
of the matrix null space and row space , column space Give a basis for the Problem 2:
= A
Find a basis for the null-space of the matrix Problem 3:
= A
= A a) Find a basis for the column space of Problem 4:
)A b) Find a basis for the perpendicular space Col( ⊥
)A c) Find a basis for Conull(
= B Let Problem 5:
: the column space, the row space, the null space andB Find a basis for the four fundamental spaces of
B the co-null space (the null space of the transpose T
Given the system of equations Problem 6:
c = z + y + x 1
(^2) c = z 2 + y 2 + x
c = z 3 + y 3 + x 3
= c~ a) For what values of
c 1
c 2
c 3
does the system have a solution?
, how many are there?~c b) If there exists a solution for a given
c) Find the basis for the co-null space of the matrix associated to the system of equations above.
d) What is the relation between your answers to part a) and c)?
R :L 5 matrix and × is a 3 A Problem 7: 5
R →
3
has rank 3. A. Suppose thatv~ · A ) =~v(L is defined by
?L a) What is the dimension of the kernel of
?L b) What is the dimension of the range of
Explain your answers in terms of how you would find basis of these spaces if the matrix of
A were given!
= A Let Problem 8:
A of the matrix Reduced Row Echelon form a) Give the
A of the matrix null-space b) Find a basis for the
A of the matrix column space c) Find a basis for the
)?A(C ) and the column spaceA( N of the null–space dimension d) What is the
, and explain your answer:False or True e) Answer
= xA~ The equation ~
has a solution for every vector b ~
R ∈ b 3
Linear Transformations and Finding a Matrix Representation
given by^3 → P^3 P :L be the space of polynomials of degree 2. Show that the map^3 P Let Problem 1:
p · x − )x(p )) =x(p( L ′
)x(
p is linear. (Here, ′
).)x(p ) denotes the first derivative of the polynomialx(
R Find the matrix, in the standard basis for Problem 2: 3
, for the linear transformation
L
x
y
z
z − y − x 2
z + y 2 − x
z + 2 y + 3 x −
L b) Find the kernel of
P :L Define the linear transformation Problem 3: 3 → P 3 by
x p )) =x(p( L ′′
x p 2 − )x( ′
)x(p ) +x(
, x, x 1 { with respect to the basis L Find the matrix representing 2
P of } 3 .
Find the matrix representation for the linear transformation Problem 4:
L
[
x
y
]
[
y − x 4
y + 4 x −
]
v~ with respect to the basis 1 =
[
]
~v and 2 =
[
]
.})x cos(2 ,)x sin(2 ,)x cos( ,)x sin({ be the space of functions with basis V Let Problem 5:
by V → V :L Define the linear transformation
f ) = f( L ′′
f + ′
f 4 −
with respect to the given basis. L a) Find the matrix representing
L b) Find the kernel of
R : T Let a linear transformation Problem 6: 3
R →
3
be defined by
v( T 1 , v 2 , v 3 v ) = ( 1 v + 2 2 v + 3 v 2 , 1 v + 2 , v 2 .)
. T basis) for standard Give the matrix (in the
e{ be the vector space spanned by the functions V Let Problem 7: x
, e x 2
, e x 3
f ) = f(L be the linear transformation defined by V → V :L and let ′
. f 2 −
e{ with respect to the basis L a) Find the matrix representing x
, e x 2
, e x 3
. V of }
.L b) Find the kernel of
R :L Define the linear transformation Problem 8: 2
R →
2
= A where vA~ ) =~v(L by
[
]
~v with respect to the new basis L Find the matrix of 1 =
[
]
v~ and 2 =
[
]
Change of Basis for Linear Transformations and Similarity
R :L The linear transformation Problem 1: 2
R →
2
= A has matrix
[
]
with respect to the
R of }^2 e, ~^1 e~{ standard basis 2
with respect to the new basis L. Find the matrix of
~v 1 =
[
]
v , ~ 2 =
[
]
e~{ with respect to the standard basis A a) Find the matrix representation Problem 2: 1 e, ~ 2 R of } 2
for the
linear transformation
L
[
x
y
]
[
y − x 4
y + 4 x −
]
~v with respect to the basis L of B b) Find the matrix representation 1 =
[
]
~v and 2 =
[
]
R :L Let Problem 3: 3
R →
3
L be the linear transformation given by
x
y
z
z + 6 y 4
y 3 − x 2 −
z + y + 2 x
~e{ with respect to the standard basis L a) Find the matrix representing 1 e, ~ 2 e, ~ 3 R of } 3
with respect to the new basis L b) Use the answer to part a) to find the matrix representing
~v 1 =
~v 2 =
~v 3 =
R Given the vectors in Problem 4: 2
~u 1 =
[
]
u~ , 2 =
[
]
v~ , 1 =
[
]
v~ , 2 =
[
]
~v{ corresponding to change of basis from S a) Find the transition matrix 1 v, ~ 2 ~u{ to } 1 u, ~ 2 .}
R :L b) The linear transformation 2
R →
2
= A has a matrix representation
[
]
with respect to the
~u{ basis 1 u, ~ 2 ~v{ with respect to the basis L of B. Find the matrix representation} 1 v, ~ 2 .}
v~ For the vectors Problem 5: 1 =
v, ~ 2 =
v, ~ 3 =
}^3 e, ~^2 e, ~^1 ~e{ corresponding to the change of basis from the standard basis S a) Find the transition matrix
R of 3
~v{ to the new basis 1 v, ~ 2 v, ~ 3 .}
R :L b) Let 3
R →
3
be the linear transformation defined by
v~( L 1 v~ ) = 1 v~( , L 2 ~v · ) = 2 2 ~v( , L 3 ~v · ) = 3 3
R of }^3 e, ~^2 e, ~^1 e~{ with respect to the standard basis L Find the matrix representing 3
Eigenvalues, Eigenvectors and Eigenspaces
= A Find the eigenvalues and corresponding eigenvectors for Problem 1:
= A Let Problem 2:
= 0.^3 , λ = 0^2 , λ = 3^1 λ are A. The eigenvalues of
a) Find the eigenvectors for these eigenvalues.
S with S matrix orthogonal is symmetric. Find an A b) Note 1 −
diagonal. D = AS
= A Find the eigenvalues and eigenvectors for Problem 3:
[
]
Systems of Differential Equations
Given the differential equations with initial conditions Problem 1:
x ′
1 = (0)x ; y 4 + x 3 =
y ′
2 = (0)y ; y 3 − x 2 − =
).t(y ) andt(x Find the functions
a) Give the general solution of the differential system Problem 2:
y ′
1 (^2) y + (^1) y =
y ′
2 y 2 − = 1 y 4 + 2
y b) Give the particular solution when 1 y (0) = 3 and 2 (0) = 1.
Finding Powers, Square Roots and Exponentials of Matrices
= A For the matrix Problem 1:
[
]
S · D · S = A such that D and S 2 matrices × a) Find 2 1 −
A b) Use your answer to part a) to calculate 5
= A For Problem 2:
[
]
e find the matrix A
2 matrix.) ×. (Your answer should be a 2
= A For Problem 3:
[
]
e 2 matrix × , find the 2 tA
= A For Problem 4:
[
]
A a) Calculate 2
A ,
3
A ,
4
A ,
5
.A b) Find the eigenvalues and eigenvectors for
A c) Use your answer to part b) to calculate 10
A d) Use your answer to part b) to get a formula for n
is a positive integer. n when
