Math 18 midterm 1

Pivots

A ________ in a matrix, A, is a position in the matrix that corresponds to a row-leading 1 in the reduced row echelon form of A. Since the reduced row echelon form of A is unique, the _______ positions are uniquely determined and do not depend on whether or not row interchanges are performed in the reduction process. Also, the _______ of a row must appear to the right of the _____________ in the above row in row echelon form.

Free Variable

A __________ _________________ is one whose parameters can be anything for the system and it will still be true. Note though that whatever value you pick for the system must remain consistent.

onto

A linear transformation, T, is ________ if its range is all of its codomain, not merely a subspace. Thus, for any vector w, the equation T(x) = w has at least one solution x (is consistent). The linear transformation T is 1-to-1 if and only if the null space of its corresponding matrix has only the zero vector in its null space. Equivalently, a linear transformation is 1-to-1 if and only if its corresponding matrix has no non-pivot columns.

Basic Variables

A variable is a ______ variable if it corresponds to a pivot column. This variable is the opposite of a free variable.

Linearly independent

A vector set is considered ___________ ____________ if it has these properties. 1. It has is homogeneous equation that has only the trivial solution.

Scalar

Any number that remains constant. ie 5 is a scalar. It is always five.

linear transformation

Definition 1 Let V , W be vector spaces. A function T : V → W is said to be a linear transformation if T(au + bv) = a T(u) + b T(v) for all u, v ∈ V and all a, b ∈ IR

Parametric Representation

Find the answers to an augmented matrix or system of equations. Then write those answers in the from of x1 = ..., x2 =...,x3=... etc. This is in the ________________________ _____________.

spans

In linear algebra, the linear _____ (also called the linear hull or just _____) of a set of vectors in a vector space is the intersection of all subspaces containing that set. The linear _____ of a set of vectors is therefore a vector space.

Span

The ______ is a set of all linear combinations of v1, v2, etc. that you can have. Say you have v1= <1,3,7>, and v2=<3,2,4>. They're _____ is the plane that they describe, and each and every possible linear combination.

codomain

The __________ of a linear transformation is the vector space which contains the vectors resulting from the transformation's action. Thus, if T(v) = w, then v is a vector in the domain and w is a vector in the range, which in turn is contained in the ___________. Examples: The ___________ of the transformation T:R3→R5 is R5 The matrix A=[1,2;2,1;1,1] (three rows and two columns) induces a linear map from R2 to R3, with domain R3

The property of Linearity

The ____________________ property is the idea that if you have an equation in the form A(*x1* + *x2*) = A*x1* + A*x2* This is basically the classical distributive property.

Indexed Set

The vectors that you are working with within a problem can be referred to as an indexed set

Elementary Row Operations

There are three operations you can perform in a matrix. 1. You can multiply a row by a scalar. 2. You can add rows. 3. You can Subtract rows.

A*x* , A

This is how we talk about matrices we denote it, _____, for ____ an mxn matrix and __ in R^ n

<=>

This symbol means if and only if.

↔

This symbol means row reduces to

linearly dependent

Vectors can be called __________ ________________ if it has these properties. 1. It is a homogeneous equation that has non trivial solutions 2. It has a free variable. 3. It has the zero vector as one of it's vectors 4. It includes two vectors which are multiples of each other.

Row equivalent

When adding or subtracting rows in a matrix you end up with a new matrix. The new matrix has one new row that is the __________ _________________ of the one you replaced.

parametric vector equations, parametric vector form

When asking for a solution to a homogeneous equation, if it has a solution that is non trivial you can put it in this form __________ ___________ _________ It will look something like *x*=X3[2;1;1] If the same matrix is not homogeneous, then it's parametric eq will look like, *x*=X3[2;1;1]+[1,1,0] This form is also called, _________________ ___________ ___

homogeneous solution

When you have a non homogeneous system equations it's solution is it's unique solution plus the ___________ ____________________.

Reduced Echelon Form

You can manipulate a matrix such that it has all the properties of Echelon form, and two more properties. These two properties are: 1. The leading entry in each row is one. 2. Each leading 1 is the only non zero entry in it's column

Echelon Form

You can manipulate a matrix such that it has three properties. These properties are: 1. All nonzero rows are above any zero rows. 2. Each leading entry in a row is to the right of the leading entry in the row above it. 3. All entries in a column bellow the leading entry are zero. This form is called ___________________ ____________ if and only if it has these properties

one to one

________ ___ ______ is where a matrix transformation is has at most one solution. For homogeneous equations this means that there is no solution aside from the trivial solution

mapping

__________, a mathematical relation such that each element of a given set (the domain of the function) is associated with an element of another set (the range of the function)

Image

____________ Eggers word for range

vector

____________ is a single column matrix.

zero vector

_____________ is a special vector that has inputs which are only zero

A*x*=*b*

_____________ is the standard matrix equation.

Homogeneous Equations

______________ ___________ are equations that are in the form A*x*=*b*. They have at least 1 solution and this solution is the zero vector, or the trivial solution.

nontrivial solutions

______________ are solutions to a homogeneous equation that are not the 0 vector

Identity matrix

______________ is a special matrix in the form [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1] but it can be any mxn matrix.

trivial solutions

_______________ are solutions of homogeneous equations that are the 0 vector.

Linear Combination

_______________ is a set of vectors added or subtracted together that are only multiplied by scalars. So basically if you have the vectors *v1 and v2* then v1+3v2 is a linear combination of those vectors

Leading entry

_______________________ ________ is the first non zero value in a row in a matrix.

range

the range is also called the column space or the image of A