Exam 1

Linear Algebra Exam 1

Unit 1

Length and Dot Products

Length Definition

The length or magnitude of a vector x in $R^{m}$ is defined to be

∣∣ x ∣∣ = x_{1}^{2} + x_{2}^{2} + \cdot \cdot \cdot + x_{n}^{2} = x \cdot x

Theorem 1

For any vector x in $R^{n}$ and scalar c in $R$ we have

$∣∣ x ∣∣ = 0$ If and only if $x = 0$
$∣∣ c x ∣∣ = ∣ c ∣∣∣ x ∣∣$

Unit Vector Definition

A vector u is called a unit vector if $∣∣ u ∣∣ = 0$

Distance Definition

The distance $d (x, y)$ between vectors x and y in $R^{n}$ is defined by $d (x, y) = ∣∣ x - y ∣∣$

Dot Product Definition

The dot product of vectors x, y in $R^{n}$ is the scalar in $R$ defined by $x \cdot y = x_{1} y_{1} + x_{2} v_{2} + \dots + x_{n} y_{n}$

Matrices and Matrix Operations

Theorem 1

If A is a $m \times k$ matrix, $B$ is a $k \times n$ matrix, C is a $n \times p$ matrix, then $(A B) C = A (BC)$

Theorem 2

If A is a $m \times k$ matrix and B is a $k \times n$ matrix then $(A B)^{T} = B^{T} A^{T}$

Consequence of Linearity

Let A be a $m \times n$ matrix and b a vector in $R^{m}$ . Then exactly one of the following is true.

$A x = b$ has no solution
$A x = b$ has exactly one unique solution
$A x = b$ has an infinite number of solutions

Properties

In general: AB $\neq =$ BA

A (B + C) = A B + A C it is associative (A + B) C = A C + BC = A BC

The transpose $A^{T}$ of A is the matrix obtained by flipping the rows and columns of A

[142536]^{T} = ⎣ ⎡ 123456 ⎦ ⎤ A^{T} = A (A B)^{T} = B^{T} A^{T}

A is symmetric if $A^{T} = A$ , example::

⎣ ⎡ 123204345 ⎦ ⎤

Suppose $v_{1} an d v_{2}$

A v_{1} = [23], A v_{2} = [50] then A (3 v_{1} + 2 v_{2}) = 3 A v_{1} + 2 A v_{2} = 3 [23] + 2 [50] = [169]

Matrix-Vector Products

A x = ⎣ ⎡ a_{11} a_{21} ⋮ a_{m 1} a_{12} a_{22} ⋮ a_{m 2} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{mn} ⎦ ⎤ ⎣ ⎡ x_{1} x_{2} ⋮ x_{n} ⎦ ⎤ = x_{1} ⎣ ⎡ a_{1} 1 a_{2} 1 ⋮ a_{m 1} ⎦ ⎤ + x_{2} ⎣ ⎡ a_{1} 2 a_{2} 2 ⋮ a_{m 2} ⎦ ⎤ + \dots + x_{n} ⎣ ⎡ a_{1 n} a_{2 n} ⋮ a_{mn} ⎦ ⎤

Unit 2

Solving Linear Systems

Augmented Matrix Definition

Consider the linear system $A x = b$ where A is a $m \times n$ matrix and b is in $R^{m}$ . We call the matrix A the corresponding coefficient matrix and the $m \times (n + 1)$ matrix $[A ∣ b]$ the corresponding augmented matrix.

Elementary Row Operations Definition

The following three operations on the rows of any matrix are called elementary row operations.

Interchange any two rows.
Multiply any row by a nonzero scalar.
Add any scalar multiple of a row to another row

Row Equivalence Definition

We call two matrices with the same number of rows and columns row equivalent if there is a sequence of elementary row operations that converts one matrix into the other.

Two $m \times n$ linear systems with corresponding augmented matrices that are row equivalent have exactly the same set of solutions.

Matrix Rank and General Solution

Matrix Rank Definition

The rank of matrix A, denoted rank( $A$ ), is the number of nonzero rows in any matrix B in row echelon form that is row equivalent to A.

General Solution Definition

The general solution of a linear system is a formula for each variable (possibility containing parameters) that generates all solutions of the linear system.

Theorem 1:

If A is a $m \times n$ matrix, then the linear system $A x = b$ has a solution for all b in $R^{m}$ if and only if rank(A) = $m$ .

RREF

RREF Definition

A matrix is said to be in reduced row echelon form if

It is in row echelon form
The first nonzero entry in each row is 1.
The first nonzero entry one each row is the only nonzero entry in its column

Theorem 1

Ever matrix is row equivalent to a unique matrix in reduce row echelon form (RREF).

Homogeneous Linear System

Homogeneous Linear System Definition

A $m \times n$ linear system is called homogeneous if $b_{1} = b_{2} = \dots = b_{m} = 0$ :

a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} = b_{1} a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} = b_{2} ⋮ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{mn} x_{n} = b_{m}

Span

Span Definition

We call the set of all linear combinations of the vectors $v_{1}, v_{2}, ..., v_{n}$ in $R^{m}$ the span of the vectors $v_{1}, v_{2}, ..., v_{n}$ which we denote $s p an (v_{1}, v_{2}, ..., v_{n})$ . We say that the vectors $v_{1}, v_{2}, ..., v_{n}$ span $R^{m}$ if $s p an (v_{1}, v_{2}, ..., v_{n}) = R^{m}$

Theorem 1

Let A be a $m \times n$ matrix with column vectors $v_{1}, v_{2}, ..., v_{n}$ and b be a vector in $R^{m}$ . The linear system $A x = b$ is consistent if and only if b is in $s p an (v_{1}, v_{2}, ..., v_{n})$ .

Theorem 2

Let A be a $m \times n$ matrix with column vectors $v_{1}, v_{2}, ..., v_{n}$ , then the following three statements are equivalent (all true or false):

$r ank (A) = m$
The linear system $A x = b$ has a solution for all b in $R^{m}$
$s p an (v_{1}, v_{2}, ..., v_{n}) = R^{m}$

Linear Independence

Linear Independence Definition

The vectors $v_{1}, v_{2} ..., v_{n}$ in $R^{m}$ are called linearly independent if the homogeneous linear system $x_{1} v_{1} + x_{2} v_{2} + ... + x_{n} v_{n} = 0$ has only the trivial solution x=0 and linearly dependent if the equation has a nonzero solution.

Theorem 1

Vectors $v_{1}, v_{2}, ..., v_{n}$ in $R^{m}$ are linearly dependent if and only if one of the vectors $v_{i}$ is in $s p an (v_{1}, ..., v_{i - 1}, v_{i + 1}, ..., v_{n})$

Theorem 2

Let A be a $m \times n$ matrix with column vectors $v_{1}, v_{2}, ..., v_{n}$ . Then the following three statements are equivalent (all true or false):

$r ank (A) = n$
The homogeneous linear system $A x = 0$ has the unique solution x=0
The vectors $v_{1}, v_{2}, ..., v_{n}$ are linearly independent

Interpretation, if the number of filled rows is equal to the number of columns, all vectors are linearly dependent.

Note on Rank:

$R ank (A) = d im (C o l (A))$