Linear Algebra
1.1 SYSTEMS OF LINEAR EQUATIONS
A system of linear equations is a collection of one or more linear equations involving the same
set of variables, say, x1, x2, … , xn.
The solution of a linear system is a list s1, s2, … , sn of numbers that makes each equation in the
system true when the values s1, s2, … , sn are substituted for x1, x2, … , xn, respectively.
A system of linear equations has either
(i) exactly one solution (consistent) or
(ii) in nitely many solutions (consistent) or
(iii) no solution (inconsistent).
Elementary Row Operations:
1. (Replacement) Add to one row a multiple of another row.
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply all entries in a row by a nonzero constant. PS: never multiply a row with zero,
as this will delete information
Two Fundamental Questions (Existence and Uniqueness)
1) Is the system consistent; (i.e. does a solution exist?)
2) If a solution exists, is it unique? (i.e. is there one & only one solution?)
1.2 ROW REDUCTION AND ECHELON FORMS
Echelon Form:
1. All nonzero rows are above any rows of all zeros.
2. Each leading entry (i.e. left most nonzero 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 zero.
Reduced Echelon Form:
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its column.
THEOREM 1 (Uniqueness of The Reduced Echelon Form)
Each matrix is row-equivalent to one and only one reduced echelon matrix.
Solution:
• basic variable: any variable that corresponds to a pivot column in the augmented matrix.
• free variable: all nonbasic variables.
THEOREM 2 (Existence and Uniqueness Theorem)
1. A linear system is consistent if and only if the rightmost column of the augmented matrix is
not a pivot column, i.e., if and only if an echelon form of the augmented matrix has no row of
the form [ 0 … 0 b ] (where b is nonzero).
2. If a linear system is consistent, then the solution contains either
(i) a unique solution (when there are no free variables) or
(ii) in nitely many solutions (when there is at least one free variable).
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, 1.3 VECTOR EQUATIONS
Vector: A matrix with only one column.
The vector y de ned by y = c1v1 + c2v2 + cpvp is called a linear combination of v1 + v2 + vp using
weights c1 + c2 + cp.
Question: Determine if b is a linear combination of a1, a2, and a3.
→ nd x1, x2, x3 such that x1a1 + x2a2 + x3a3 = b.
Span{v1, v2, … , vp} = set of all linear combinations of v1, v2, … , vp.
1.4 THE MATRIX EQUATION Ax = b
If A is an m×n matrix, with columns a1, a2, … an, and if x is in Rn, then the product of A and x,
denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x
as weights.
THEOREM 3
If A is a m×n matrix, with columns a1, … an, and if b is in Rm, then the matrix equation
Ax = b has the same solution set as the vector equation x1a1 + … + x3a3 = b which, in turn, has
the same solution set as the system of linear equations whose augmented matrix is
[ a1 a2 … an b ].
The equation Ax = b has a solution if and only if b is a linear combination of the columns of A.
The columns of A = [ a1 a2 … ap ] span Rm if every vector b in Rm is a linear combination of
a1, …, ap (i.e. Span {a1, … ap} is in Rm).
THEOREM 4
Let A be an m×n matrix. Then the following statements are logically equivalent:
a. For each b in Rm, the equation Ax = b has a solution.
b. Each b in Rm is a linear combination of the columns of A.
c. The columns of A span Rm.
d. A has a pivot position in every row.
THEOREM 5
If A is an m×n matrix, u and v are vectors in Rn, and c is a scalar, then:
a. A(u + v) = Au + Av;
b. A(cu) = c(Au)
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, 1.5 SOLUTION SETS OF LINEAR SYSTEMS
Homogeneous Linear Systems
A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0.
The zero solution is often called the trivial solution. The homogeneous equation has a nontrivial
solution if the equation has at least one free variable.
Parametric Vector Form
The equation of the form x = su + tv is called a parametric vector equation of the plane.
Whenever a solution set is described explicitly as a linear combination of vectors, we say that the
solution is in parametric vector form.
Solutions of Non-homogeneous Systems
For a non-homogeneous linear system with many solutions, the general solution can be written
in parametric vector form as a single arbitrary solution + the parametric vector form of the solution
set of the corresponding homogeneous system.
Writing a Solution Set in Parametric Vector Form
1. Row reduce the augmented matrix to reduced echelon form.
2. Express each basic variable in terms of any free variables appearing in an equation.
3. Write a typical solution x as a vector whose entries depend on the free variables, if any.
4. Decompose x into a linear combination of vectors (with numeric entries) using the free
variables as parameters (coe cients).
1.6 APPLICATIONS OF LINEAR SYSTEMS:
BALANCING CHEMICAL EQUATIONS
(?) C3H8 + (?) O2 → (?) CO2 + (?) H2O (x1) C3H8 + (x2) O2 → (x3) CO2 + (x4) H2O
ffi
1.1 SYSTEMS OF LINEAR EQUATIONS
A system of linear equations is a collection of one or more linear equations involving the same
set of variables, say, x1, x2, … , xn.
The solution of a linear system is a list s1, s2, … , sn of numbers that makes each equation in the
system true when the values s1, s2, … , sn are substituted for x1, x2, … , xn, respectively.
A system of linear equations has either
(i) exactly one solution (consistent) or
(ii) in nitely many solutions (consistent) or
(iii) no solution (inconsistent).
Elementary Row Operations:
1. (Replacement) Add to one row a multiple of another row.
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply all entries in a row by a nonzero constant. PS: never multiply a row with zero,
as this will delete information
Two Fundamental Questions (Existence and Uniqueness)
1) Is the system consistent; (i.e. does a solution exist?)
2) If a solution exists, is it unique? (i.e. is there one & only one solution?)
1.2 ROW REDUCTION AND ECHELON FORMS
Echelon Form:
1. All nonzero rows are above any rows of all zeros.
2. Each leading entry (i.e. left most nonzero 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 zero.
Reduced Echelon Form:
4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its column.
THEOREM 1 (Uniqueness of The Reduced Echelon Form)
Each matrix is row-equivalent to one and only one reduced echelon matrix.
Solution:
• basic variable: any variable that corresponds to a pivot column in the augmented matrix.
• free variable: all nonbasic variables.
THEOREM 2 (Existence and Uniqueness Theorem)
1. A linear system is consistent if and only if the rightmost column of the augmented matrix is
not a pivot column, i.e., if and only if an echelon form of the augmented matrix has no row of
the form [ 0 … 0 b ] (where b is nonzero).
2. If a linear system is consistent, then the solution contains either
(i) a unique solution (when there are no free variables) or
(ii) in nitely many solutions (when there is at least one free variable).
fi fi
, 1.3 VECTOR EQUATIONS
Vector: A matrix with only one column.
The vector y de ned by y = c1v1 + c2v2 + cpvp is called a linear combination of v1 + v2 + vp using
weights c1 + c2 + cp.
Question: Determine if b is a linear combination of a1, a2, and a3.
→ nd x1, x2, x3 such that x1a1 + x2a2 + x3a3 = b.
Span{v1, v2, … , vp} = set of all linear combinations of v1, v2, … , vp.
1.4 THE MATRIX EQUATION Ax = b
If A is an m×n matrix, with columns a1, a2, … an, and if x is in Rn, then the product of A and x,
denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x
as weights.
THEOREM 3
If A is a m×n matrix, with columns a1, … an, and if b is in Rm, then the matrix equation
Ax = b has the same solution set as the vector equation x1a1 + … + x3a3 = b which, in turn, has
the same solution set as the system of linear equations whose augmented matrix is
[ a1 a2 … an b ].
The equation Ax = b has a solution if and only if b is a linear combination of the columns of A.
The columns of A = [ a1 a2 … ap ] span Rm if every vector b in Rm is a linear combination of
a1, …, ap (i.e. Span {a1, … ap} is in Rm).
THEOREM 4
Let A be an m×n matrix. Then the following statements are logically equivalent:
a. For each b in Rm, the equation Ax = b has a solution.
b. Each b in Rm is a linear combination of the columns of A.
c. The columns of A span Rm.
d. A has a pivot position in every row.
THEOREM 5
If A is an m×n matrix, u and v are vectors in Rn, and c is a scalar, then:
a. A(u + v) = Au + Av;
b. A(cu) = c(Au)
fi fi
, 1.5 SOLUTION SETS OF LINEAR SYSTEMS
Homogeneous Linear Systems
A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0.
The zero solution is often called the trivial solution. The homogeneous equation has a nontrivial
solution if the equation has at least one free variable.
Parametric Vector Form
The equation of the form x = su + tv is called a parametric vector equation of the plane.
Whenever a solution set is described explicitly as a linear combination of vectors, we say that the
solution is in parametric vector form.
Solutions of Non-homogeneous Systems
For a non-homogeneous linear system with many solutions, the general solution can be written
in parametric vector form as a single arbitrary solution + the parametric vector form of the solution
set of the corresponding homogeneous system.
Writing a Solution Set in Parametric Vector Form
1. Row reduce the augmented matrix to reduced echelon form.
2. Express each basic variable in terms of any free variables appearing in an equation.
3. Write a typical solution x as a vector whose entries depend on the free variables, if any.
4. Decompose x into a linear combination of vectors (with numeric entries) using the free
variables as parameters (coe cients).
1.6 APPLICATIONS OF LINEAR SYSTEMS:
BALANCING CHEMICAL EQUATIONS
(?) C3H8 + (?) O2 → (?) CO2 + (?) H2O (x1) C3H8 + (x2) O2 → (x3) CO2 + (x4) H2O
ffi
