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Mathematics

MATH 136

Eddie Dupont

Winter

Description

Linear Algebra
Definition:
Linear Algebra (L.A.) is the study of vector spaces, where vector space is a set of stuff where each
elements of the set is called a vector.
Note: Vectors are not necessarily a ‘column of numbers’, rather, ‘column of numbers’ are an example of
vector in a particular vector space.
Vector Spaces:
Definition:
For , let⃗ [ ]with for , the set of all possible ⃗ is called. An individual ⃗
with is called a vector in .
Note: It will be sometimes useful to visualize the vectors t⃗ as points in a Cartesian system
(especially and ).
The two operations of vectors are called vector addition, and scalar multiplication, which you are free to
make it totally whacky. For now we create the ‘intuitive’ operations as follows.
Definition:
Let ⃗ [ ] ⃗ [ ] ⃗ ⃗ , and c is scalar.
Vector Addition (V.A.) in
⃗ ⃗ [ ] [ ] [ ]
Scalar Multiplication (S.M.) in
⃗ [ ] Axioms: Vector Spaces
Let ⃗ ⃗ ⃗⃗⃗
1. ⃗ ⃗
2. ( ⃗ ⃗ ) ⃗⃗⃗ ⃗ ⃗ ⃗⃗⃗
3. ⃗ ⃗ ⃗ ⃗
4. ⃗⃗ ⃗ ⃗⃗ ⃗ ⃗
5. ⃗ ( ⃗) ⃗ ( ⃗) ⃗⃗
6. ⃗
7. ( ⃗) ⃗
8. ⃗ ⃗⃗⃗ ⃗
9. ( ⃗ ⃗) ⃗ ⃗
10. ⃗ ⃗
If the above properties hold, then the set is called a vector space, and the elements are called vectors.
Note:
1. Proposition 2 (associative of addition), allows us to add as many vectors in as we want and still
get something in , and as a bonus, we can drop the parentheses.
2. If V is a vector space, then these properties hold.
a. ⃗ ⃗⃗ ⃗
b. ⃗ ⃗ ⃗
Linear Combination
Definition:
Let ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ be a set of vectors in vector space . The object
⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ belongs to is called a linear combination of
vector space .
Span
Definition:
Let ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ be a set of vectors in vector space . We define all possible linear
combinations of B the span of B and written as ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
. We say that B is a spanning set of or is spanned by B.
Behavior of Span A span of a set of vectors gives all possible linear combination of the vectors. This often lead to
redundancies.
Linear Dependence
Definition:
Let ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ be a set of vector in vector space V.
⃗⃗
B is called linear dependent if there is a non-trivial solution to ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ .
B is called linearly independent if the only solution to ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗ is the trivial
case.
Note: A trivial solution means .
Theorem:
1. Span is a subspace of .
2. ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ if and only if ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
3. A set ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ is linearly dependent if and only if ⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
4. If contains ⃗⃗then it is is linearly dependent.
Definition:
⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
Let . B is called a basis for if the vectors in is linearly
independent and .
⃗⃗
Note: Basis for { } is defined to be empty set, .
A standard basis in is ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ where we denote by ⃗⃗⃗⃗ the vector with I in the i-th component
and 0 elsewhere.
Theorem
Let ⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ be a basis for . Any vector ⃗⃗⃗⃗ can be written as a unique linear
combination of the vectors in B.
Subspaces
Definition:
A subset is a subspace of if all of the properties of the Axiom of Vector Space hold, but with
replaced by . Namely ⃗ ⃗ and , then we need:
⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗
⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ Subspace Test
Let S be a non-empty subset of and ⃗ ⃗ . If is closed under vector addition and cosed
under scalar multiplication, then is a subspace of .
Theorem:
If ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗ , then ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗ is a subspace of .
Dot Product (Scalar product/ Standard inner product)
Let ⃗ [ ] ⃗ [ ]
1. We define the dot product as ⃗ ⃗ .
2. We define the norm (or length) of ⃗ as ⃗ √ ⃗ ⃗ √∑ .
3. We define the angle between ⃗ ⃗ to satisfy ⃗ ⃗ ⃗ ⃗ .
Note: The dot product is SCALAR.
Finding a basis for Subspace S
1. Use the description of the set find a general vector in S.
2. Identify a set of vectors that spans S.
3. If they are linear independent, then we are done. Otherwise remove redundant vectors until we
get a linearly independent set.
Theorem:
If ⃗ ⃗ , we define the angle between ⃗ ⃗ to satisfy ⃗ ⃗ ⃗ ⃗ .
If ⃗ ⃗ ⃗⃗, then ⃗ ⃗ are said to be orthogonal.
Properties of Dot Product
Let ⃗ ⃗ ⃗ and , then
1. ⃗ ⃗ and ⃗ ⃗ ⃗ ⃗⃗
2. ⃗ ⃗ ⃗ ⃗
3. ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗
Note: Proofs are based on definition.
Properties of Norm
If ⃗ ⃗ , and , then ⃗⃗
1. ⃗ and ⃗ ⃗
2. ⃗ | | ⃗
3. |⃗ ⃗ | ⃗ ⃗ (Cauchy-Schawrz inequality)
4. ⃗ ⃗ ⃗ ⃗ (Triangle inequality)
5. ⃗ ⃗ ⃗ ⃗ (Reverse triangle inequality)
Cross Product
The cross product is an construction that is useful in many physical and geometric plane. It is created
by forcing a vector ⃗⃗ [ ] to be orthogonal to the plane formed by two linearly independent vectors
⃗ ⃗⃗⃗ .
⃗ [ ] ⃗⃗⃗⃗⃗ [ ] ⃗ ⃗⃗⃗⃗⃗ [ ]
Note:
⃗ ⃗⃗⃗ is a vector
By definition, ⃗ ⃗ ⃗⃗⃗ ⃗⃗⃗ ⃗ ⃗⃗⃗
⃗ ⃗⃗⃗ ⃗ ⃗⃗⃗, non-commutative.
Properties of Cross Product
1. If ⃗ ⃗ ⃗⃗⃗ ⃗ ⃗ ⃗⃗⃗
2. ⃗ ⃗⃗⃗ ⃗⃗⃗ ⃗
⃗⃗
3. ⃗ ⃗
4. ⃗ ⃗⃗⃗ ⃗⃗ ⃗ ⃗⃗⃗ ⃗⃗ ⃗ ⃗⃗⃗
5. ⃗ ⃗⃗⃗ ⃗ ⃗⃗⃗⃗ ⃗⃗⃗ ⃗ ⃗
6. ⃗ ⃗⃗⃗ ⃗ ⃗⃗⃗
7. ⃗ ⃗⃗⃗ ⃗ ⃗⃗⃗ | | Equation of Plane
Given 2 linearly independent vectors ⃗ ⃗⃗⃗ and a point ⃗⃗ , then the equation of the plane is:
1. Vector form
⃗ ⃗⃗ ⃗ ⃗⃗⃗ .
2. Scalar form
⃗ ⃗⃗ ⃗ ⃗⃗
Where ⃗⃗ [ ] ⃗ ⃗⃗⃗ and ⃗ ⃗⃗.
Projection& Perpendicular
Projection onto Vectors
The projection of a vector ⃗⃗ onto ⃗ is the amount of corresponding
vector in the ⃗. The formula is given by
⃗⃗ ⃗
⃗⃗⃗ ⃗ ⃗
The perpendicular of vector ⃗⃗ on ⃗ is the perpendicular distance
between the point ⃗⃗ and ⃗⃗⃗. It is given by the formula:
⃗⃗ ⃗⃗ ⃗⃗ Figure : ⃗⃗⃗⃗ , ⃗⃗⃗⃗⃗ .
⃗⃗ ⃗⃗
Projection onto Planes
Definition:
Let ⃗⃗ and P be a plane in with normal
vector ⃗⃗⃗⃗.
We define ⃗⃗ ⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗.
Figure : Projection of a vector onto a plane. System of Linear Equation
Definition:
1. Let be unknowns, and be known, then
[Comments]
is a system of m linear equations with n unknowns. Each line of the system is a linear equation
in standard form.
2. A vector ⃗ [ ] is called a solution to \(≧⊔≦)/ if each line of \(≧⊔≦)/ is true when
the are replaced with . All such ⃗ form a solution set.
3. If \(≧⊔≦)/ has no solution, it is call inconsistent. If \(≧⊔≦)/ has at least one solution, it is
call consistent.
Theorem
The system \(≧⊔≦)/ has either:
No solution.
Exactly one solution, or
Infinitely many solutions.
Definition:
1. Two systems are equivalent if they have the same solution set.
2. The coefficient matrix of \(≧⊔≦)/ is the rectangle array [ ].
3. The augmented matrix of \(≧⊔≦)/ is [ | ]
Note: We often write the coefficient matrices ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗ when ⃗⃗⃗⃗ [ ] and
⃗⃗ [ ] so that the augmented matrix is written as [ |⃗] where A is an matrix and [ |⃗] is an
matrix. Reduced Row Echelon Form (RREF)
An augmented matrix is said to be in RREF if it satisfies all the conditions below:
1. All zero rows are placed at the bottom.
2. The first non-zero entry in a non-zero row is 1 (Called the leading one).
3. In any two successive non-zero rows, the leading one is in the lower row occurs further to the
right than the leading one in the row above, and
4. Every columns with a leading one have zero’s above and below it throughout the column.
To arrive at the RREF, we perform any one of the 3 Elementary Row Operations (ERO):
1. Multiply a row by a non-zero scalar.
2. Adding a multiple of one row to another row.
3. Swapping two rows.
Note: When converting matrix to its RREF form, you should indicate the row operations at each step.
Furthermore you are allowed to perform more than one ERO on different rows in a single step.
Definition:
1. Two matrices are row equivalent if one is obtained from the other by ERO.
2. If R is the RREF of A in |⃗⃗ (with unknowns ) and the j column of R does not
[ ]
contain a leading one, then is called a free variable.
Note: If at any point in the process, we see a row like the following | (a.k.a. the
‘Nasty’ Row), then the system will be inconsistent.
Definition:
The rank of a matrix (written as rank(A)) is the number of leading ones in the RREF of A.
Fact: If A is an mxn matrix, then .
Theorem:
(The awesome theorem!!!)
Let A be an matrix, ⃗⃗ and construct [ | ]⃗ (i.e. m equations with n unknowns), then
1. If [ | ]⃗, then the system is inconsistent.
2. If |⃗⃗ is consistent, then there are free variables.
3. |⃗⃗ is consistent ⃗⃗ . Homogeneous System
Definition:
⃗⃗
The system [ | ] is called a Homogeneous system.
Note that regardless of Rank(A), the vector ⃗ ⃗⃗is always a solution called the trivial solution.
If is an matrix, and Rank(A) = n (number of columns), then by the awesome theorem part 2,
there are free variables, and so only the trivial solution exists.
Theorem:
The solution set of a homogeneous system of m equations in n unknowns is a subspace of called the
solution space.
Note: Prove by subspace test. (There’s an easier proof using matrix algebra.)
This means that given two solutions ⃗ ⃗ to a homogeneous system [ | ]⃗, then ⃗ ⃗ is also a
solution, so is ⃗.
Note: This is generally not true for system [ | ] with ⃗⃗ ⃗.
Theorem:
Let , then
1. Let [ | ]⃗ be a consistent system with RREF [ | ] , if Rank(A) = Kn, then C is linearly dependent.
2. If and are both bases for , then k=n. (Dimension theorem)
Span-Linear dependence theorem
If is n-dimensional and n>0, then
A set of more than n vectors is linearly dependent.
A set of less than n vectors cannot span
A set of n vectors in is linearly independent if and only if it spans .
Extend to a basis
If is n-dimensional and ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ is linearly independent with k

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