SUMMARY OF MATH 1600
Note: The following list is intended as a study guide for the final
exam. It is a continuation of the study guide for the midterm. It does
not claim to be a comprehensive list. You should also study class notes,
the textbook, recommended problems and MapleTA problems.
1. Vectors
1.1. Introduction.
(1) Know how to manipulate n-vectors via addition, subtraction
and scalar multiplication using rules of Theorem 1.1.
(2) Understand the geometric definition of addition and scalar multiplication of vectors and how to obtain vectors defined with
initial point A and terminal point B.
(3) Find linear combinations of vectors.
(4) Do arithmetic in Zm and Znm .
(5) Solve equations in Zm and Znm or determine if this is not possible.
1.2. Dot Product.
(1) Compute the dot product of 2 vectors in Rn or Znm and the
length of a vector in Rn . Find the distance between two vectors
in Rn .
(2) Know and be able to compute with the dot product and length
using properties of Theorem 1.2 and 1.3.
(3) Know how to normalize a non-zero vector in Rn .
(4) Know and be able to apply The Cauchy Schwarz Inequality and
the Triangle inequality (Theorems 1.4 and 1.5).
(5) Be able to compute the angle between two vectors in Rn and
be able to determine when two vectors are orthogonal.
(6) Know and be able to apply Pythagorus’ Theorem (Theorem
1.6).
(7) Be able to compute the projection of a vector onto another
non-zero vector.
1.3. Lines and Planes.
(1) Determine the vector, parametric, normal and general equations
of a line in R2 and be able to convert between any two of these
forms.
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SUMMARY OF MATH 1600
(2) Determine the vector and parametric forms of a line in R3 .
(3) Determine the vector, parametric, normal and general equations
of a plane in R3 and be able to convert between any two of these
forms.
(4) Find the distance between a point and a line in R2 or R3 and
find the closest point on the line to the given point.
(5) Find the distance between a point and a plane in R3 and find
the closest point on the plane to the given point.
(6) Find the angle between two non-parallel planes.
(7) Find the distance between two parallel planes or lines.
1.4. Code Vectors.
(1) Find a missing digit in a code vector given a check vector in
Znm . Do this in the specific cases of UPC codes and ISBN-10
codes.
(2) Determine whether or not a single digit error or a specified
transposition error will be detected in a code vector with given
check vector.
2. Systems of Linear Equations
2.1. Introduction.
(1) Find the solution to a system of linear equations algebraically
by backwards or forwards substitution or geometrically by interpreting the system as the intersection of lines in R2 or planes
in R3 .
2.2. Direct Methods for Solving Linear Equations.
(1) Row reduce a matrix into row echelon form or reduced row
echelon form using a series of elementary row operations.
(2) Use Gaussian elimination to solve a system of linear equations.
(3) Use Gauss-Jordan elimination to solve a system of linear equations.
(4) Know the Rank Theorem and its implications.
(5) Find the rank of a matrix.
(6) Find the intersection of 2 non-parallel planes.
(7) Find, if possible, the intersection of 2 lines in R3 .
(8) Know Theorem 2.3: that a homogeneous system with more
variables than equations has a non-zero solution and its implications.
(9) Solve linear systems of equations over Zp .
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2.3. Spanning Sets and Linear Independence.
(1) Determine when a given vector is in the span of a set of vectors.
(2) Be able to use Theorem 2.4: i.e. b ∈ Span{a1 , . . . , an } if and
only if [A|b] is consistent where ai is the ith column of A.
(3) Determine when a subset of Rn spans Rn or not.
(4) Describe the span of a subset of Rn geometrically if n = 2, 3.
(5) Determine when a subset of Rn or Znp is linearly independent
or dependent. If dependent, find a linear dependence relation.
(6) Use Theorem 2.6 to check for linear dependence/independence
m
(or Zm
of a set of vectors in Rm (or Zm
p )
p ) {a1 , . . . , an } ⊂ R
is linearly independent if and only if the homogeneous system
Ax = 0 has only the zero solution. If it isP
linearly dependent,
a dependence relation can be found using ni=1 xi ai = 0 if and
only if Ax = 0. Here A is the m × n matrix A = [a1 , . . . , an ].
(7) Know the relationships between linear independence and rank.
A set of vectors {a1 , . . . an } in Rm is linearly independent if and
only if rank(A) = n where A = [a1 , . . . , an ] is the m × n matrix
with {a1 , . . . , an } as columns.
(8) Know the relationship between spanning and rank. A set of
vectors {a1 , . . . , an } ⊆ Rm spans Rm if and only if rank(A) = m
where A = [a1 , . . . , an ] is the m × n matrix with {a1 , . . . , an } as
columns.
(9) In general, rank(A) ≤ min{m, n} where A is m by n. Know the
implications of this for systems of linear equations.
(10) Know that any set of m vectors in Rn must be linearly dependent if m > n.
2.4. Applications. We skipped this section.
3. Matrices
3.1. Matrix Operations.
(1) Know all matrix terms and notation: rows, columns,diagonal
entries, diagonal matrix, scalar matrix, identity matrix, square
matrix, zero matrix.
(2) Know when 2 matrices are equal.
(3) Compute sums, differences, negatives, and scalar multiples of
matrices, if defined.
(4) Compute products of matrices if defined.
(5) Compute powers of square matrices and know properties of
powers.
(6) Compute the product of two partitioned matrices.
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SUMMARY OF MATH 1600
(7) Be able to manipulate various partitions of matrix products.
Suppose A is m × n and B is n × p where ai , bi are the ith
columns of A, B and Ai , Bi are the ith rows of A, B respectively.
If ei ∈ Rm , then eTi A = Ai = ith row of A.
If ej ∈ Rn , then Aej = aP
j = jth column of A.
Row i of AB is Ai B or nj=1 aij Bj .
P
Column j of AB is Abj or ni=1 bij ai .
These are all different special cases of partitioning matrices
to compute products.
(8) Compute the transpose of a matrix and determine when a matrix is symmetric.
3.2. Matrix Algebra.
(1) Know and be able to use the properties of matrix addition and
scalar multiplication of Theorem 3.2.
(2) Determine when a matrix is a linear combination of a set of
other matrices.
(3) Describe the span of a set of matrices.
(4) Determine when a set of matrices is linearly independent or
dependent and if dependent, find a linear dependence relation.
(5) Know and be able to use the properties of Matrix multiplication
given by Theorem 3.3.
(6) Know and be able to use the properties of transposes given by
Theorem 3.4.
(7) Know that matrix multiplication isn’t commutative and its consequences. Know that many multiplicative facts that work in
the real numbers do not work for matrices, such as AB = 0
doesn’t imply A = 0 or B = 0 and its consequences.
(8) Know how to test when a matrix is symmetric, and how to obtain symmetric matrices from arbitrary matrices as in Theorem
3.5.
3.3. The Inverse of a Matrix.
(1) Know the definition of an inverse of a matrix and how to check
when a matrix is invertible, and how to check when a matrix B
is an inverse of a matrix A (and hence A−1 = B since inverses
are unique by Theorem 3.6).
(2) Know how to determine whether a 2 by 2 matrix is invertible,
and if so, find its inverse by the formula in Theorem 3.8.
(3) Find the unique solution to Ax = b if A is invertible using the
formula x = A−1 b (Theorem 3.7).
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(4) Know and be able to use the properties of inverses with respect
to inverses, products, scalar multiples, transposes, and powers
as in Theorem 3.9.
(5) Be able to find the elementary matrix of size m obtained from
any of the three types of elementary row operations (i.e. do the
elementary row operation to Im ).
(6) Know and be able to use Theorem 3.10: Multiplying an m by n
matrix A on the left by an elementary matrix E corresponding
to an elementary row operation performs the same elementary
row operation on A.
(7) Find the inverse of an elementary matrix using Theorem 3.11:
The inverse of an elementary matrix corresponding to an elementary row operation is the elementary matrix corresponding
to the inverse of that elementary row operation.
(8) Be able to write an invertible matrix as a product of elementary
matrices.
(9) Find an elementary matrix which performs an elementary row
operation.
(10) Know and be able to use the Fundamental Theorem of Invertible
Matrices: Version I (Theorem 3.12).
(11) Find the inverse of an n × n matrix A, if it exists, by row
reducing [A|In ] to [In |A−1 ]. (This works by Theorem 3.14). A
isn’t invertible if rank(A) < n where A is n × n.
(12) Check that an n × n matrix A is invertible by just checking
AB = In for some n × n matrix B. (Theorem 3.13).
(13) Find the inverse of a matrix with coefficients in Zp , p prime.
3.4. This section was not covered.
3.5. Subspaces, Basis, Dimension, and Rank.
(1) Know the definition of a subspace of Rn , and be able to check
whether a subset of Rn is a subspace by checking the three axioms or if it is not, provide an explicit numerical counterexample
that shows that it fails one of the three axioms.
(2) Know that the span of a finite set of vectors of Rn is a subspace
(Theorem 3.19).
(3) Know the definitions of the row space, column space and null
space of an m × n real matrix A and that Row(A), Null(A) are
subspaces of Rn (latter from 3.21) and Col(A) is a subspace of
Rm .
(4) Know the definition of a basis B of a subspace S of Rn and be
able to verify that a subset B of S is a basis of S.
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SUMMARY OF MATH 1600
(5) Know that row equivalent matrices have the same row space.
Use this and the fact that RREF(A) has row space basis given
by its non-zero rows to find a basis for Row(A). That is, a basis
of Row(A) is given by the non-zero rows of its RREF.
(6) Find a basis for Col(A) by using {ap(1) , . . . , ap(r) } where p(1) <
· · · < p(r) are the pivot columns of the RREF of A. (i.e. a
basis for Col(A) is given by the columns of A corresponding to
pivot columns of RREF(A).)
(7) Find a basis for Null(A) using the fact that row equivalent matrices have the same row space so that Null(A) = Null(R) where
R =RREF(A). Solve the system Rx = 0 by setting all the nonpivot variables to parameters, writing all the pivot variables in
terms of the parameters and then after substituting back into x,
expressing the solutions as linear combinations of n − rank(A)
vectors, one corresponding to each non-pivot variable. These
n − rank(A) vectors are a basis of Null(A).
(8) Know the basis theorem, that any two bases for a subspace S
of Rn , have the same number k of vectors. Then S is a kdimensional vector space.
(9) Determine the dimension of any subspace of Rn .
(10) In particular, determine the dimension of subspaces related
to a matrix A: dim(Col(A)) = dim(Row(A)) = rank(A) and
nullity(A) = dim(Null(A)) = n − rank(A) if A is m × n.
(11) Know and be able to use the Rank theorem: rank(A)+nullity(A) =
n if A is m × n.
(12) Know that rank(A) = rank(AT ) (Theorem 3.25).
(13) Know and be able to use The Fundamental Theorem of Invertible Matrices: Version 2.
(14) Find a basis for a subspace given as a spanning set of a finite
set of vectors in Rn .
(15) Determine whether or not a set of vectors in Rn (resp Znp ) forms
a basis of Rn (resp. Znp )
(16) Compute the rank and nullity of matrices over R or Zp .
(17) Determine whether a vector is in Row(A), Col(A) or Null(A)
for a given matrix A.
(18) Know how to find the possible values of rank and nullity of a
matrix of a given size, and to make conclusions about the set
of columns or rows of the matrix.
(19) Show that a vector w ∈ Rn is in Span(B) for a basis B of a
subspace S of Rn . Determine its coordinate vector [w]B in this
case.
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3.6. Introduction to Linear Transformations.
(1) Know the definition of a linear transformation and be able to
prove that a map is a linear transformation or disprove that
it is a linear transformation using an explicit numerical counterexample.
(2) Know that the map TA : Rn → Rm , TA (x) = Ax is a linear
transformation for any real m × n matrix A (Theorem 3.30).
(3) Know in fact that any linear transformation T : Rn → Rm is
of the form T = TA for a unique real m × n matrix A. In fact
T = TA if and only if A = [T ] where [T ] = [T (e1 ), . . . , T (en )] is
the standard matrix of the linear transformation T . (Theorem
3.31).
(4) Determine the standard matrix of a linear transformation T :
Rn → Rm .
(5) For each type of linear transformation given by geometry in R2 .
find its standard matrix. That is, find the matrix of Rθ : R2 →
R2 , rotation by θ radians counterclockwise from the positive
x-axis (Example 3.58). Find the matrix of projection Pl onto
a line l through the origin in R2 . (Note that Pl (x) = projd (x)
where d is the direction vector of the line through the origin
in R2 ). Find the matrix of reflection Rl in a line l through the
origin in R2 . (Note that Fl (x) = 2projd (x) − x where d is the
direction vector of the line through the origin in R2 ).
(6) Find the composite of linear transformations T : Rn → Rm ,
S : Rm → Rp , given by S ◦ T : Rn → Rp , (S ◦ T )(v) = S(T (v)).
From Theorem 3.32, S ◦ T is also a linear transformation with
standard matrix [S ◦ T ] = [S][T ].
(7) Find the composite of 2 linear transformations directly and via
Theorem 3.32.
(8) Find the matrix of a composite of linear transformations of R2
given by geometric descriptions.
(9) Know how to check whether a linear transformation is invertible
and verify that another linear transformation is its (unique)
inverse.
(10) A linear transformation T : Rn → Rn is invertible if and only if
its standard matrix [T ] is invertible. In this case [T −1 ] = [T ]−1
(Theorem 3.33 plus class). Use this formula to find the inverse
of a linear transformation.
4. Eigenvalues and Eigenvectors
4.1. Introduction.
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SUMMARY OF MATH 1600
(1) Know and be able to use the definition of an eigenvector and
eigenvalue of a square matrix. Be able to verify whether a given
vector is an eigenvector and to find a corresponding eigenvalue.
(2) Find the eigenvalues of a 2 by 2 matrix over R, C or Zp .
(3) Find a basis for each eigenspace of a 2 by 2 matrix over R, C
or Zp .
(4) For a 2 by 2 matrix which is the standard matrix of a linear
transformation given geometrically, find its eigenvalues and a
basis for each eigenspace.
4.2. Determinants.
(1) Find the determinant of a square matrix using the cofactor or
Laplace expansion along any row or column (Theorem 4.1).
(2) Know and be able to use the fact that the determinant of a triangular matrix is the product of the diagonal entries (Theorem
4.2).
(3) Know and be able to use the properties of determinants coming
from elementary row and column operations (Theorem 4.3).
(4) Use row reduction, keeping track of elementary row operations
and their effects on determinant plus the result on the determinant of a triangular matrix, to compute the determinant of a
matrix.
(5) Know the determinants of the three types of elementary matrices (Theorem 4.4).
(6) Know and be able to use the fact that a square matrix is invertible if and only if its determinant is non-zero.
(7) Know and be able to use facts about determinants: The determinant of a scalar multiple (Theorem 4.7), the determinant of
a product (Theorem 4.8), the determinant of the inverse of an
invertible matrix (Theorem 4.9), the determinant of a transpose
(Theorem 4.10). Use those rules to find the determinant of a
matrix formed by the above matrix operations.
(8) Use elementary row and column operations to deduce the determinant of a matrix from that of a given matrix.
(9) Use determinant properties and equations satisfied by matrices
to find possible values of determinants.
(10) Use Cramer’s rule (Theorem 4.11) to solve a system of linear
equations given by Ax = b where A is an invertible square
matrix.
(11) Use the adjoint formula to find the inverse of an invertible matrix.
Cross Product (Exploration after 4.2:Determinants)
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(1) Know how to compute the cross product of two vectors in R3
using the determinant.
(2) Know properties of cross product from Exercise 3 of Exploration
(after Determinants 4.2).
(3) Determine the area of a parallelogram or triangle determined
by 2 vectors in the plane using the cross product.
(4) Determine the volume of a parallelopiped determined by 3 vectors in R3 .
(5) Find the equation of a plane passing through 3 points.
4.3. Eigenvalues and eigenvectors of n by n matrices.
(1) Find the characteristic polynomial of a square matrix.
(2) Find the eigenvalues of a square matrix by finding the roots of
its characteristic polynomial.
(3) Find a basis for each eigenspace for each eigenvalue of a square
matrix.
(4) Find the algebraic and geometric multiplicities of each of the
eigenvalues of a square matrix.
(5) Know and be able to use the fact that the eigenvalues of a
triangular matrix are the diagonal entries.
(6) Know yet another equivalent condition to the invertibility of a
square matrix A, that 0 is not an eigenvalue of A leading to
Theorem 4.17.
(7) Know and be able to use the formulas for the eigenvalues of
powers and inverses of matrices with respect to a given eigenvector for the matrix. (Theorem 4.18).
(8) Know and be able to use the fact that a set of eigenvectors
for a square matrix A corresponding to distinct eigenvalues is
linearly independent (Theorem 4.20).
(9) Find Ak x for a diagonalisable matrix A.
(10) Find conditions on eigenvalues of matrices satisfying equations.
4.4. Similarity and Diagonalisation.
(1) Know the definition of similar matrices and be able to use it.
(2) Know that similarity satisfies the properties of Theorem 4.21
(is an equivalence relation on square matrices).
(3) Know that similar matrices share many properties: determinant, rank, characteristic polynomial, eigenvalues.
(4) Determine when square matrices are not similar by computing
ranks, determinants, characteristic polynomials.
(5) Know that if matrices are similar, so are their powers and even
negative powers if the matrices are invertible.
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SUMMARY OF MATH 1600
(6) Know the definition of diagonalisable matrix.
(7) Be able to use Theorem 4.23 to diagonalise a matrix if it is
possible.
(8) Know that the union of bases of the eigenspaces corresponding to the distinct eigenspaces is always linearly independent
(Theorem 4.24).
(9) Know that an n by n matrix with n distinct eigenvalues is always
diagonalisable (Theorem 4.25).
(10) Know that the geometric multiplicity of an eigenvalue of a
square matrix is always less than its algebraic multiplicity.
(11) Know and be able to use the Diagonalisation Theorem, which
determines precisely when a square matrix with characteristic
polynomial which factors into linear factors is diagonalisable.
This is true if and only if the algebraic multiplicity is equal to
the geometric multiplicity of each distinct eigenvalue. In this
case a basis of Rn consisting of eigenvectors for the n by n
real matrix A is given by the union of bases of the eigenspaces
corresponding to distinct eigenvalues.
(12) Compute the powers of a diagonalisable matrix A using its diagonalisation.
5. Orthogonality
5.1. Orthogonality in Rn .
(1) Know the definition of an orthogonal (respectively orthonormal)
set in Rn and be able to recognise them.
(2) Know and be able to use the fact that an orthogonal set of
non-zero vectors in Rn is linearly independent.
(3) Know the definition of and be able to recognise an orthogonal
or orthonormal basis of Rn .
(4) If w ∈ Span(B) where B = {v1 , . . . , vk } is an orthogonal subset
of non-zero vectors in Rn , then find w as a linear combination
of the vectors in B using the formula in Theorem 5.2. Equivalently, find the coordinates of w with respect to B.
(5) Know the definition of an orthogonal square real matrix and
its equivalent descriptions in Theorem 5.4, Theorem 5.5, and
Theorem 5.7. Be able to determine whether or not a matrix is
orthogonal and, if so, find its inverse.
(6) Know that a square real matrix is orthogonal if and only it
preserves lengths if and only if it preserves dot products (and
hence angles) (Theorem 5.6).
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(7) Know that the inverse of an orthogonal matrix is orthogonal
and the product of orthogonal matrices is orthogonal. Know
that orthogonal matrices have determinant ±1 and eigenvalues
of complex length 1. (Theorem 5.8). Note : the length of
a complex number is also called its magnitude. The complex
length of a real number is its absolute value.
(8) Know how to do arithmetic in complex numbers, find complex
conjugates of complex numbers, and lengths of complex numbers. Know that the complex conjugate of a product is the
product of the complex conjugates and the length of a product of complex numbers is the product of their lengths. (Note:
length of a complex number is also called its magnitude).
(9) Know that orthogonal 2 by 2 matrices are either matrices of
rotations or reflections and recognise them as such.
5.2. Orthogonal Complements and Orthogonal Projections.
(1) Know the definition of the orthogonal complement of a subspace
of Rn and that it is a subspace of Rn . Be able to find a basis of
this subspace.
(2) Know that the null space of a matrix and the row space of a
matrix are orthogonal complements and that the column space
of a matrix and the null space of its transpose are orthogonal
complements. (Theorem 5.10). Find bases of each of the 4
fundamental subspaces of a matrix A: i.e. Row(A), Col(A),
Null(A), and Null(AT ).
(3) Know properties of orthogonal complements given by Theorem
5.9.
(4) Find the orthogonal projection of a vector in Rn onto a subspace
W of Rn . Note that the definition of orthogonal projection
given in the text requires an orthogonal basis of W but does
not depend on the choice of orthogonal basis. Find also the
component of a vector orthogonal to W .
(5) Know and be able to use the Orthogonal Decomposition theorem (Theorem 5.11). Find the orthogonal decomposition of a
vector in Rn with respect to a subspace W of Rn .
5.3. The Gram Schmidt Process and the QR factorisation.
(1) Note that we will not cover the QR factorisation of a matrix.
(2) Construct an orthogonal basis of a subspace of Rn from an
arbitrary basis via the Gram-Schmidt Process (Theorem 5.15).
(3) Find an orthogonal basis of Rn that contains a given orthogonal
subset.
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SUMMARY OF MATH 1600
5.4. Orthogonal Diagonalisation of Symmetric Matrices.
(1) Know the definition of an orthogonally diagonalisable matrix.
(2) Be able to orthogonally diagonalise a real symmetric n by n
matrix A. That is, find its eigenvalues and an orthonormal
basis for each distinct eigenspace. Put the orthonormal basis
B = {q1 , . . . , qn } of Rn consisting of eigenvectors for A into
the columns of a matrix Q = [q1 , . . . , qn ]. Then QT AQ =
diag(λ1 , . . . , λn ) where Aqi = λi qi . Also find its spectral decomposition A = λ1 q1 qT1 + · · · + λn qn qTn .
(3) Know that a real square matrix is orthogonally diagonalisable
if and only if it is symmetric (Theorem 5.17 and 5.20).
(4) Know that if A is a real symmetric matrix, then any two eigenvectors corresponding to distinct eigenvalues are orthogonal
(Theorem 5.19) and the eigenvalues of a real symmetric matrix are real (Theorem 5.18).
(5) Use the spectral decomposition to determine a symmetric matrix with a specified set of orthogonal eigenvectors corresponding to specified eigenvectors.
(6) Use the spectral decomposition theorem to show that certain
combinations of orthogonally diagonalisable matrices are orthogonally diagonalisable.