Linear Algebra mid term study

Consistent

Consistent

If a linear system is _________, then the solution set contains either (i) a unique solution, when there are no free variables,

or

(ii) infinitely many solutions, when there is at least one free variable.

USING ROW REDUCTION TO SOLVE A LINEAR SYSTEM

1

ELEMENTARY ROW OPERATIONS

T

Vector

A matrix with only one column is called a column _____, or simply a ______. Examples of _____ with two entries are

What does R stand for in for example R²

The set of all vectors with two entries is denoted by R². The R stands for the real numbers that appear as entries (like rows) in the vectors, and the exponent 2 indicates that each vector contains two entries.

Two vectors in R2 are equal if and only if their corresponding entries (rows) are equal.

Vectors in R³

3 x 1 column matrices with three entries. They are represented geometrically by points in a three-dimensional coordinate space, with arrows from the origin sometimes included for visual clarity.

Vectors in R^n

T

Properties of Vectors in R^n

T

Linear Combination

T

Span

The set of all possible vectors that you can reach with a linear combination of a given pair of vectors is called the ____ of those two vectors.

How do you figure out if a vector is in the span of 2 vectors?

T

THE MATRIX EQUATION

Ax = b

Theorem 3

T

true or false: The equation Ax = b has a solution if and only if b is a linear combination of the columns of A.

true

T

identity matrix

a square matrix in which all the elements of diagonals are one, and all other elements are zeros. It is denoted by the notation “I_n” or simply “I”. where the N stands for the R^n its in (number of rows)

T

General Solution

y = p+h
where p is the particular solution and h is the homogenous solution.

Homogenous

T

Trivial Solution

If there are no free variables and Ax = 0 you have a ______

Non-trivial solution

T

Example of a General Solution

Linearly IN - DEPENDENT

has only trivial solutions

Linearly DEPENDENT

Has non trivial solution

True or False: If a set S = {v1, ... , v p} in contains the zero vector, then the set is linearly dependent.

True

Why is a set linearly dependent if there are more columns than rows

If there are more columns than rows that means there are less pivots than the amount of columns which simultaneously means there isn’t a pivot for at least one of the columns resulting in a free variable.

True or False: If a set S {v1, ... , v p} in contains the zero vector, then the set is linearly dependent.

True

Linearly dependent requirements

If a vector is a linear combination of another vector they are linearly dependent.

Ex: if a vector is a scalar multiple of another
v1= cv2

Transformation

Image

For x in R^n the vector T (x) in is called the image of x (under the action of T).

basically in this example T(x) is transforming x which is in R² (domain) into R³ (codomain) by multiplying x with A

Elementary vectors

How can you tell if there is more than one x whose image under T is b?

If you have free variables after solving the matrix

Linear Transformation

Contractions and Dilations

Counter Clockwise rotation matrix (plug in -1*theta for clockwise)

How do you find Domain and Codomain

Onto

Has free variable

Range

Range example

Horizontal Contraction/ expansion

Vertical Contraction / Expansion

True or False: Onto is related to consistency

true

One to one

no free variables only one trivial solution

True or False: If there are more columns than rows it is onto

True

basically what a is saying is that if any vector b in R^m can be made up of the columns of A multiplied by some scalar
b=c1​a1​+c2​a2​+⋯+cn​an​ then its onto

Some things you can do with matrixes

Product Size of multiplying matrices

More stuff you can do with Matrices

Things you can do With transposes

Invertible

Finding the inverse of a 2×2 matrix

Determinant of a 2×2

True or False: If A is an invertible n x n matrix, then for each b in, the equation Ax = b has the unique solution x = A^-1 b.

True

properties of invertible Matrices

how do you take the inverse of a matrix bigger than 2×2

you basically set it up so it’s side by side with the Identity matrix and you try to row reduce until the identity matrix is on the opposite side then you have your inverse matrix. If at any time you get a free variable or a row of 0s then the matrix in un invertible.

Invertible Matrix theorem

Invertible Transformations

cool

LU Factorization

Made up of the upper and lower triangular matrices which when multiplied gives you the original matrix

Hoe do you solve a LU factorization problem for x?

Step 1: First you do [L :b] or life is beautiful to get your y vector.

Step 2: next you do [U:y] to get your x vector.

LU decomposition

turning a regular matrix into LU form Requires you to first

Step 1: put the matrix into row echelon form make sure to keep track of each time you eliminate the rest of a column. you have to use ONLY REPLACEMENT. This will give you the upper triangular matrix.
Step 2: Place the columns in the L format by taking the pivot column right before reducing the rest of the column to 0s and scale them so 1 is a pivot. This will give you the lower triangular matrix.

Subspace

Both Nul Spaces and Col Spaces are subspaces

Column Space

Ignore the hand writing

Nul Space

Basically all the x values such that Ax=0

Subspace example

example of a non subspace

A span is always a subspace

Remember a vector is in the span if it can be made up of linear combinations of {v1,v2,…vp}

so the 0 vector is in the span because you can jus

Span goonified

Trivial Subspaces

basically vectors are always in the span

and any vector in R^n is also in the subspace because If you are in R^n you have a 0 vector if you add two nx1 vectors you still get an nx1 vector and samething if you multiply an nx1 vector by a scalar c

When does Col(A) not = R³?

when there is a row in the row reduced matrix that is all 0s and the augmented matrix in that row does not equal 0.

How do you determine if the x vector is in Nul A

If you Ax=0

Basis

nxn can be a base in R^n because if for example in R³ you have a 3×2 matrix there are only 2 pivots, while a 3×4 matrix definitely has a free variable. But if you have a 4×3 matrix its still possible for it to be a base. at the most a matrix can be a base for R^n if the size is mxn.

How do you determine if a matrix is a base for R^n?

Row reduce and if there are free variables it isn’t other wise it is.

How to find the basis for Col(A)

Row reduce the matrix and only the PIVOT COLUMNS are the basis for Col(A)

How to find the basis for Nul(A)

Row reduce and find all the solutions for x in Ax=0. Write out your general solution and the vectors in that solution make up your basis.

Subspace of Nul(A)

The null space of a mxn matrix A in Ax = 0 is the subspace of R^n.

Vectors in the basis of Nul(A)

are the same amount as how many free variables there are.

If you end up getting no free variables when solving for the basis of Nul(A)

then you get a trivial basis or 0 vector

Coordinate Vector

How do you find the coordinate vector of a Matrix

you take the span vectors and make an augmented matrix with the x vector and then row reduce.

Dimension

the amount of vectors in the basis

Rank

Basically just the dimension of column space (the amount of vectors in the basis of the column space of A)

The Rank Theorem

Dimension of Nul(A)

is equal to the amount of free variables.

Invertible Matrix Theorem Continued

Determinant

Calculating the Determinant with Cofactor expansion

Basically you pick a row or column and for each value in that row or column you cross out the values in the same row and column and the remaining numbers form a determinant matrix of lower size and at the end you add them all up. MAKE SURE to multiply each of your factors by -1^(row +column) depending on their position in the matrix.

this is good if there are row or columns in the matrix that have 0s

finding the determinant using triangular matrix

basically just reducing the matrix so it creates a diagonal which can then be multiplied to give the determinant. THERE ARE RULES FOR THIS METHOD HOWEVER

Row operation rules when using the triangular matrix method to find the determinant.

True

True

Properties for determinants

Cramers Rule

How to find detA_i(b)

basically just replace the i’th column in (A_i) with b and calculate the determinant.

How to Use Cramers Rule

An Inverse formula

a has to be an nxn matrix

Area of a parallelogram

Example of Area of Parallelogram