All linear algebra Flashcards
What is a linear equation?
An equation that can be written in the form a1x1 + a2x2 + … + anxn = b
What is a system of linear equations?
A collection of one or more linear equations involving the same variables
What is a solution of a system of linear equations?
A list (s1, s2, … sn) of numbers that makes each equation a true statement when the values s1,…,sn are substituted for x1,…,xn, respectively.
What is the solution set of a linear system?
The set of all possible solutions
When are two linear systems equivalent?
When they have the same solution set
What types of solutions can a linear system have?
1 - No solution
2 - One solution
3 - Infinitely many solutions
When is a linear system consistent?
When it has either one solution or many solutions
When is a linear system inconsistent?
When it has no solutions
What is a matrix?
A rectangular array
What is a coefficient matrix?
A matrix containing the coefficients of a linear system
What is an augmented matrix?
The coefficient matrix with an added column containing constants from the right sides of a system of linear equations
What are the elementary row operations?
1- Replacement - Replace one row by the some of itself and a multiple of another row
2- Interchange - Interchange two rows
3- Scaling - Multiply all entries in a row by a nonzero constant
When are two matrices row equivalent?
If there is a sequence of elementary row operations that transforms one matrix into the other.
When do two systems have the same solution set?
When the augmented matrices of the two systems are row equivalent.
When is a rectangular matrix in echelon form?
1- All nonzero rows are above any rows of all zeroes
2- Each leading 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 zeroes
When is a rectangular matrix in reduced echelon form?
When it is in echelon form, AND:
1- The leading entry in each nonzero row is 1
2- Each leading 1 is the only nonzero entry in its column
What is Theorem 1?
Uniqueness of the Reduced Echelon Form:
Each matrix is row equivalent to one and only one reduced echelon matrix.
What is a pivot position in a matrix A?
A location in A that corresponds to a leading 1 in the reduced echelon form of A.
What is a pivot column of matrix A?
A column of A that contains a pivot position
What is a pivot?
A nonzero number in a pivot position that is used as needed to create zeroes via row operations
What is the forward phase of the row reduction algorithm?
The steps required to bring it into echelon form
What is the backward phase of the row reduction algorithm?
The steps required to bring it from echelon form to reduced echelon form
What are basic variables?
Variables that correspond to pivot columns in a matrix
What are free variables?
Variables for which you are free to choose any value
What is Theorem 2?
Existence and Uniqueness Theorem:
A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column (coefficient matrix has no row all 0s with the corresponding augmented column being nonzero).
If a linear system is consistent, then the solution set contains either (1) a unique solution, when there are no free variables (2) infinitely many solutions, when there is at least one free variable.
What is a column vector (or vector)?
A matrix with only one column
When are two vectors in R^2 equal?
When and only when their corresponding entries are equal
What is the parallelogram rule for addition?
If u and v in R^2 are represented as points in the plane, then u+v corresponds to the fourth vertex of the parallelogram whose other vertices are u, 0, and v.
What is the zero vector?
The vector whose entries are all zero
What are the algebraic properties of of R^n?
1- u + v = v + u 2- (u + v) + w = u + (v + w) 3- u + 0 = 0 + u = u 4- u + (-u) = -u + u = 0 5- c(u+v) = cu + cv 6- (c+d)u = cu + du 7- c(du) = (cd)u 8- 1u = u
What is a linear combination?
Given vectors v1 v2 … vp in R^n and given scalars c1 c2 … cp, the vector y defined by y = c1v1 + … + cpvp
What are the weights of a linear combination?
The constant coefficients on each vector
How do you tell if a linear combination is valid?
A vector equation x1a1 + x2a2 + … + xnan = b has the same solution set as the linear system whose augmented matrix is [a1 a2 … an b]. b can be generated by a linear combination of a1 … an iff there exists a solution to the linear system corresponding to that augmented matrix.
What is Span(v1 … vp)?
The collection of all vectors that can be written in the form c1v1 + c2v2 + … cpvp with c1 .. cp scalars.
What is the set of all linear combinations of v1 … vp denoted by Span(v1 … vp) called?
The subset of R^n spanned (or generated) by v1 … vp
What is the product of A and x Ax?
The linear combination of the columns of A using the corresponding entries in x as weights.
What is an equation of the form Ax=b called?
A matrix equation
What are the two points of Theorem 3?
If A is an mxn matrix with columns a1…an and if b is in R^m, the matrix equation Ax=b:
- Has the same solution set as the vector equation x1a1 + x2a2 + … + xnan = b
- Has the same solution set as the system of linear equations whose augmented matrix is [a1 a2 … an b]
When does the equation Ax=b have a solution?
Iff b is a linear combination of the columns of A
What are the 4 logically equivalent points of Theorem 4?
- For each b in R^m, the equation Ax=b has a solution
- Each b in R^m is a linear combination of the columns of A
- The columns of A span R^m
- A has a pivot position in every row
What is the row-vector rule for computing Ax?
If the product Ax is defined, then the ith entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector x.
What are the two points of Theorem 5?
If A is an mxn matrix, u and v are vectors in R^n, and c is a scalar, then:
- A(u + v) = Au + Av
- A(cu) = c(Au)
When is a system of linear equations homogeneous?
If it can be written in the form Ax=0.
How many solutions does a system Ax=0 have?
At least one
Under what circumstances does the homogeneous equation Ax=0 have a nontrivial solution?
If and only if the equation has at least one free variable
How can the solution set of a homogeneous equation Ax=0 always be expressed, explicitly?
Span(v1 … vp)
Under what circumstances is the solution set of Ax=0 a line through the origin?
If Ax=0 has only one free variable
Under what circumstances is the solution set of Ax=0 a plane through the origin?
If Ax=0 has two or more free variables
What equation describes the solution set of Ax=0 in parametric vector form?
x = tv
What equation describes the solution set of Ax=b in parametric vector form?
x = p + tv
What is the equation of the line through p parallel to v?
x = p + tv
What is the solution set of Ax=b?
A geometric figure (line, plane) through p parellel to the solution set of Ax=0
What is Theorem 6?
If Ax=b is consistent for some given b, and p is a solution, then the solution set of Ax=b is the set of all vectors of the form w = p + vh, where vh is any solution of the homogeneous equation Ax=0
