Midterm Flashcards
Consistant
Has more than 0 solutions
Uniqueness
Has 1 solution
Define m and n for Amxn
m is rows, n is columns
True or false. Each matrix has a unique RREF. As such, two matrices cannot be wrote equivalent to the same RREF
False. A given matrix in RREF can be equivalent to an infinite amount of matrices, whilst any given matrix can only have one RREF
True or false. An RREF is an REF.
True. The rules for an RREF satisfy all conditions for an REF
Basic Variables
Variables which are not free
True or false. Rough reduction to the REF is enough to determine if a solution exists and is unique.
True, although you may not know the exact solution.
True or false. The span of any set of vectors is at least a line.
Fase. Consider the case of a set that contains only the zero vector.
What is the name of the equation Ax= b
The matrix equation
What type of matrix is A in the matrix equation Ax =b
Coefficient matrix
Identify the vectors in the homogenous equation Ax=0
X and the 0 vector
T or F. For any given matrix A, the homogeneous equation has 1 solution.
False, for any given matrix A the homogeneous equation has at least one solution (the trivial solution)
What is the geometric interpretation of the following parametric vector equations?
X=tv
X=su+tv
Line, plane
The solution set for Ax=0 is a line. What can we say about the solution set for Ax=b
It is either a line or nonexistent
Write the following solution set in parametric vector form and determine its geometric interpretation.
X1= 3+4x3
X2=5+7x3
X3 is free
[3;5;0]=x3 [4,7,1]
T or F. A set of three vectors in R3 may or may not be linearly dependent and spans at minimum a line
False. They may all be the zero vector
If Ax =0 has the trivial solution what is the relation of the columns of A?
We cannot say because Ax=0 always has the trivial solution
True or False. The range and codomain of a transformation are interchangeable terms that mean the same thing
False. The codomain is the space of the transformations of the columns of A. It is possible that the range is the whole codomain, however it is not necessarily true in all scenarios
In the transformation given by Ax=b the equation is consistent, identify wether the columns of A and the vectors x and b exist in the range codomain and domain respectively.
The columns of A exist in the range (within the codomain), the vector x exists in the domain and the vector b exists in the range (within the codomain)
What is the name of a matrix that represents a linear transformation T?
The standard matrix
T or F A transformation of Rn to Rm is said to be one to one if each b in Rn has only one solution
False. Each b in Rm can have a max of one solution, it can have no solutions
T or F. A transformation from Rn to Rm is said to be onto if the range is the codomain
True
Identify with which fundamental questions of linear algebra the term onto and one to one correspond with respectively
Onto->existence, one to one->uniqueness
T(e1) = (1,3,6), T(e2) = (2,8,0) Find the standard matrix domain and codomain of the transformation, and give the geometric description of the range
Standard matrix A =[1 2; 3 8; 6 0] domain is R2 and codomain is R3 the range is a plane in R3
The transformation T of R2 -> R2 first reflects points through the horizontal x axis then reflects points through the origin. What is the standard matrix of the transformation?
A = [-1 0; 01]
Rotate the vector [2;3] 90 degrees ccw and reflect through the horizontal x axis.
[-3;-2]
List the diagonal entries if matrix A [123;456;789]
1,5,9
True of False Aand B commute. This the addition A+B is defined
Fase. Consider the case where A is 3x4 and B is 4x3
True or False. The matrix multiplication A(At) is always defined
True. If A is mxn, At is nxm
True or False. Given a matrix A, At is always defined
True
What terms refer ti a non invertible and invertible matrix respectively
Singular and non singular
True or False A is am mxn matrix. A is invertible.
False. A must be an nxn matrix in order to be invertible
Are all nxn matrixes invertible
No
What is the determinant of the matrix A = [12;-1-2]
0
Is [12;-1-2] invertible
No
What are three properties of a subspace
It must contain the zero vector, be closed under addition and be closed under scalar multiplication
Define column space and null space
Column space is the set of all linear combinations of the columns of A, the range of the transformation represented by A the span of the vectors that form the column of A these are all the column space. The null space is the set of all solutions to the equation Ax=0
In the equation Ax=b what is the representative of the column space and the null space respectively
The column space is all bs whereas the null space is all xs.
Describe the process for determining a basis for the column space of a matrix A.
Find the REF then identify pivot columns. Corresponding columns from the original matrix are the basis for the column space of a.
What are the two properties that are basis must have?
The vectors in the basis must be literally independent and span the subspace
Describe the process for determining a basis for the nail space of a matrix A.
Compute the solution set to Ax=0 form an augmented matrix, find RREF, write parametric form.
How many ways can I get in vector be written with respect to a basis?
Only one. Consider Cartesian coordinates.
The number of vectors in a basis is referred to as what?
The dimension
What is the term for the dimension of the column space of matrix A
Rank
What is the term for the dimension of the nail space of matrix A?
Nullity
Solution set
Set of all possible solutions of a linear system
Equivalent
Two linear systems are called equivalent if they have the same solution set
Inconsistent
No solutions
Three basic operations to simplify linear systems
1) replace one equation by the sum of itself and a multiple of another equation
2) interchange two equations
3) multiply all the terms in an equation by a non-0 constant
Elementary row operations
One replace
Two interchange
Three scaling
Note: they are reversible
Row Equivalent
Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. They will have the same solution set.
Is the system consistent?
Does at least one solution exist?
Is the solution unique?
If a solution exists, is it the only one?
Leading Entry
The left most non-0 entry in a non-0 row
Non-zero row or column
Any column/row in a matrix that contains at least one and non-0 entry
Three properties of REF
1) all non-0 rows are above any rows of all zeros.
2) each lead entry of a row is in a column to the right of the leading entry of the row above it
3) all entries in column below leading entry are zero.
Properties of RREF
1) the leading entry in each row is one
2) each leading one is the only non-0 entry in its column.
Is REf unique
No
Is RREF unique
Yes
Row Echelon form of A
If a matrix a is row equivalent to an echelon matrix you, then U is a row, equivalent form of A
The row reduced echelon form of A
If you is in a row, reduced echelon form the reduced echelon form of A
Pivot position
The location in matrix that corresponds to a leading one in the RREF form
Pivot column
A column of a matrix that contains a pivot position
Basic variables
Variables which correspond to pivot columns in the matrix
Free variables
Variables from non-pivot columns
In a parametric description, what are free variables?
Free variables, active parameters
Column vector, vector
Matrix with only one column
R2
The set of all vectors with two entries
Zero vector
Vector whose entries are all zero, denoted by 0
Asking if B is in the span of {v1…vp}
If x1v1+…+xpvp = b
Homogeneous
Ax=0
Ax= 0 facts
- Always has the trivial solution
- Has non trivial solution if more than 1 free variable.
Domain
Rn
Codomain
Rm
One to one
If each b in Rm is the image of at most one x in Rn. Is a uniqueness question. Needs pivot position in every row
Determinant if 2x2 matrix
ad-bc, can’t equal 0
Three properties of subspaces in Rn
1) Zero vector is in subspace
2) For each u and v, the sum of u+v is in subspace
3) For each u in H and each scalar c, the vector cu is in H
