LINEAR ALGEBRA EXAM STUDY SET
QUESTIONS AND ANSWERS
Three Types of Systems - Answer-A. one unique solution (consistent) where all the
variables are leading
B. infinitely man solutions (consistent) where there is at least one free variable
C. no solution (inconsistent) where 0 = 1.
rank of a matrix - Answer-The number of leading 1's in rref(A), denoted rank(A)
Number of equations vs. number of unknowns (Theorem 1.3.3) - Answer-A. If a linear
system has exactly one solution, then there must be at least as many equations as
there are variables (m ≤ n ).
B. A linear system with fewer equations than unknowns (n < m) has either no solutions
or infinitely many solutions.
Definition 1.3.9: Linear combinations - Answer-A vector b in Rn is called a linear
combination of the vectors v1 , . . . , vm in Rn if there exists scalars x1,...,xm such that:
b = x1v1 + ... + xmvm.
Theorem 1.3.10 Algebraic Rules For Ax - Answer-If A is an n x m matrix, x and y are
vectors in Rn, and k is a scalar then:
A) A(x + y) = Ax + Ay
B) A(kx) = k(Ax)
Definition 2.1.1: Linear transformations - Answer-A function T from Rm to Rn is called a
linear transformation if there exists an n × m matrix A such that T(x) = Ax for all x in the
vector space Rm.
Theorem 2.1.3: Linear Transformation Requirements - Answer-A transformation T from
Rm to Rn is linear if (and only if)
a. T(v + w) = T(v) + T(w), for all vectors v and w in Rm, and
b. T(kv) = kT(v), for all vectors v in Rm and all scalars k.
Definition 2.1.4: Distribution vectors and transition matrices - Answer-A vector x in Rn is
said to be a distribution vector if its components add up to 1 and all the components are
positive or zero.
A square matrix A is said to be a transition matrix if all its columns are distribution
vectors.
If A is a transition matrix and x is a distribution vector, then Ax will be a distribution
vector as well.
Scaling Matrix - Answer-For any positive constant k, the matrix [k 0 ; 0 k] defines a
scaling by k
, Orthogonal Projections Formulas - Answer-If w is a nonzero vector parallel to L, then:
projL(x) = (x • w / (w • w))w.
If u is a unit vector parallel to L then:
projL(x) = (x • u)u.
Orthogonal Projections Matrices - Answer-P = 1/(w₁² + w₂²)[w₁² w₁w₂ ; w₁w₂ w₂²] = [u₁²
u₁u₂ ; u₁u₂ u₂²].
Reflections Formula - Answer-refL(x) = 2projL(x) - x = 2(x•u)u - x
Reflection Matrix - Answer-[a b ; b -a] where a² + b² = 1.
Rotations Matrix - Answer-The matrix of a counterclockwise rotation in R² through an
angle θ is
[cosθ -sinθ ; sinθ cosθ].
Note this matrix is of the form [a -b ; b a], where a² + b² = 1. Any matrix of this form
represents a rotation.
Rotations Combined with A Scaling - Answer-A matrix of the form [a -b ; b a] represents
a rotation combined with a scaling.
Matrix Multiplication Rules - Answer-a. Let B be an n × p matrix and A a q × m matrix.
The product BA is defined if (and only if) p = q.
b. T(x) = B(Ax) = (BA)x, for all x in the vector space Rm. The product BA is an n x m
matrix.
The column of the matrix product - Answer-Let B be an n × p matrix and A a p × m
matrix with columns v₁, v₂, ... vm. Then the product BA is:
BA = B[v₁, v₂, ... vm] = [Bv₁, Bv₂, ... Bvm].
To find BA, we can simply multiply B by the columns of A and combine the resulting
vectors.
Matrix Multiplication is noncummutative - Answer-AB ≠ BA, in general. However, at
times it does happen that AB = BA; then we say that the matrices A and B commute.
Multiplying with the identitiy matrix - Answer-For an n x m matrix A, AIm = InA = A.
Theorem 2.3.6: Multiplication is Associative - Answer-(AB)C = A(BC) = ABC.
Theorem 2.3.7: Distributive property for matrices - Answer-If A and B are n × p matrices,
and C and D are p × m matrices, then
A(C + D) = AC + AD, and
(A + B)C = AC + BC.
QUESTIONS AND ANSWERS
Three Types of Systems - Answer-A. one unique solution (consistent) where all the
variables are leading
B. infinitely man solutions (consistent) where there is at least one free variable
C. no solution (inconsistent) where 0 = 1.
rank of a matrix - Answer-The number of leading 1's in rref(A), denoted rank(A)
Number of equations vs. number of unknowns (Theorem 1.3.3) - Answer-A. If a linear
system has exactly one solution, then there must be at least as many equations as
there are variables (m ≤ n ).
B. A linear system with fewer equations than unknowns (n < m) has either no solutions
or infinitely many solutions.
Definition 1.3.9: Linear combinations - Answer-A vector b in Rn is called a linear
combination of the vectors v1 , . . . , vm in Rn if there exists scalars x1,...,xm such that:
b = x1v1 + ... + xmvm.
Theorem 1.3.10 Algebraic Rules For Ax - Answer-If A is an n x m matrix, x and y are
vectors in Rn, and k is a scalar then:
A) A(x + y) = Ax + Ay
B) A(kx) = k(Ax)
Definition 2.1.1: Linear transformations - Answer-A function T from Rm to Rn is called a
linear transformation if there exists an n × m matrix A such that T(x) = Ax for all x in the
vector space Rm.
Theorem 2.1.3: Linear Transformation Requirements - Answer-A transformation T from
Rm to Rn is linear if (and only if)
a. T(v + w) = T(v) + T(w), for all vectors v and w in Rm, and
b. T(kv) = kT(v), for all vectors v in Rm and all scalars k.
Definition 2.1.4: Distribution vectors and transition matrices - Answer-A vector x in Rn is
said to be a distribution vector if its components add up to 1 and all the components are
positive or zero.
A square matrix A is said to be a transition matrix if all its columns are distribution
vectors.
If A is a transition matrix and x is a distribution vector, then Ax will be a distribution
vector as well.
Scaling Matrix - Answer-For any positive constant k, the matrix [k 0 ; 0 k] defines a
scaling by k
, Orthogonal Projections Formulas - Answer-If w is a nonzero vector parallel to L, then:
projL(x) = (x • w / (w • w))w.
If u is a unit vector parallel to L then:
projL(x) = (x • u)u.
Orthogonal Projections Matrices - Answer-P = 1/(w₁² + w₂²)[w₁² w₁w₂ ; w₁w₂ w₂²] = [u₁²
u₁u₂ ; u₁u₂ u₂²].
Reflections Formula - Answer-refL(x) = 2projL(x) - x = 2(x•u)u - x
Reflection Matrix - Answer-[a b ; b -a] where a² + b² = 1.
Rotations Matrix - Answer-The matrix of a counterclockwise rotation in R² through an
angle θ is
[cosθ -sinθ ; sinθ cosθ].
Note this matrix is of the form [a -b ; b a], where a² + b² = 1. Any matrix of this form
represents a rotation.
Rotations Combined with A Scaling - Answer-A matrix of the form [a -b ; b a] represents
a rotation combined with a scaling.
Matrix Multiplication Rules - Answer-a. Let B be an n × p matrix and A a q × m matrix.
The product BA is defined if (and only if) p = q.
b. T(x) = B(Ax) = (BA)x, for all x in the vector space Rm. The product BA is an n x m
matrix.
The column of the matrix product - Answer-Let B be an n × p matrix and A a p × m
matrix with columns v₁, v₂, ... vm. Then the product BA is:
BA = B[v₁, v₂, ... vm] = [Bv₁, Bv₂, ... Bvm].
To find BA, we can simply multiply B by the columns of A and combine the resulting
vectors.
Matrix Multiplication is noncummutative - Answer-AB ≠ BA, in general. However, at
times it does happen that AB = BA; then we say that the matrices A and B commute.
Multiplying with the identitiy matrix - Answer-For an n x m matrix A, AIm = InA = A.
Theorem 2.3.6: Multiplication is Associative - Answer-(AB)C = A(BC) = ABC.
Theorem 2.3.7: Distributive property for matrices - Answer-If A and B are n × p matrices,
and C and D are p × m matrices, then
A(C + D) = AC + AD, and
(A + B)C = AC + BC.
