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MATH 416 Midterm 1
| Term | Definition |
|---|---|
| Linear equation | LE in the variables x1,...,xn is an equation of the form a1x1+...+anxn=b where ai is in the complex numbers with -1<i<n+1 and b is also in the complex numbers |
| Linear system | A linear system is a collection of one or more linear equations involving the same variables |
| Solution set | The set of all possible solutions to a system of linear equations |
| If there is a system of linear equations | It has either one, zero, or infinitely many solutions |
| Consistent system | A system is consistent if it has at least one solution |
| Inconsistent system | A system is inconsistent if it has no solutions |
| The elementary operations that do not change the solution of a system of linear equations | Interchange of two rows, scaling by a non-zero constant, replacing one row by the sum of itself and a multiple of another row |
| Row echelon form | A matrix is in row echelon form if all non-zero rows are above any rows of all zeros, each leading entry of a row is in a column to the right of the leading entry of the row above it, and all entries in a column below a leading entry are zero |
| Reduced row echelon form | A matrix is in reduced row echelon form if it is in row echelon form, each leading entry in a non-zero row is 1, and each leading 1 is the only non-zero entry in its column. Matrices reduced to RREF are unique |
| Pivots | Leading entries of each of the rows in REF |
| Pivot position | The position corresponding to the leading entry in REF |
| Pivot column | Column of the matrix that contains aA pivot position |
| Fields | A field is a set F with two distinct elements 0,1 included in F and two operations * and +, both from FxF to F |
| Field properties / requirements | Fields must satisfy the following properties: commutativity, associativity, identities (0 and 1), additive and multiplicative inverses, and the distributive property |
| F^n | F^n is the set of all lists of length n of elements in F |
| Vector addition for F^n | Adds corresponding coordinates |
| Scalar multiplication for F^n | Multiplies each coordinate by a field element |
| Vector space | A vector space over a field F is a set V together with vector addition and scalar multiplication |
| Vector space properties / requirements | Vector spaces must satisfy the following properties: commutativity of addition, associativity of addition and multiplication, an additive identity, an additive inverse, a multiplicative identity, and the distributive property |
| Smallest vector space | The zero vector |
| Some other vector space properties | A vector space has a unique additive identity, every u in V has a unique additive inverse, 0*u=0v for all u in V, a*0v=0v for all a in F, -1*u=-u if -1 is the additive inverse of F |
| Subspace | A subset of vector space V is a subspace if it forms a subspace over the same field F with the same vector addition and scalar multiplication |
| Conditions for a subspace | The only conditions to verify a subspace are that it includes the additive identity, is closed under addition, and is closed under scalar multiplication |
| Sum of subspaces | The set of all possible vectors formed by adding one vector from each subspace |
| The sum of all subspaces in V is a subspace of V | The sum of all subspaces in V is also the smallest subspace of V that contains all subspaces |
| Direct sum | A sum of subspaces is called a direct subspace if every vector from each subspace can be written in only one way |
| Conditions for a direct sum | A sum of subspaces is a direct sum iff the only way to write 0v as a sum of vectors is by picking each vector to be 0v |
| Direct sum of two subspaces | A sum of two subspaces is a direct sum iff the only element in their intersection is 0v |
| Linear combination | A linear combination of vectors is another vector formed by multiplying the other vectors by a field element and adding them |
| Span | Span is the set of all linear combinations of a list of vectors. If the span of a list of vectors is equal to V, we say the list spans V |
| Span of the empty set and other properties | The span of the empty set is defined to be 0v. It is declared to be linearly independent |
| If there exists a list of vectors in a vector space V | Then its span is the smallest subspace of V containing all vectors in the list |
| Finite-dimensional | A vector space is called finite-dimensional if some finite list of vectors in it spans the space |
| Infinite-dimensional | A vector space is called infinite-dimensional if it is not finite-dimensional |
| Linear independence | A list of vectors is linearly independent if the only way to use them to form 0v is by multiplying them all by 0 |
| Linear dependence | A list of vectors is linearly dependent if it is not linearly independent |
| Linear Dependence Lemma | If there is a linearly dependent list of vectors in V, there exists one vector in the span of all the other vectors, and that vector can be removed without changing the span of all the other vectors |
| Length of linearly independent and spanning lists | The length of a linearly independent list must be less than or equal to the length of the spanning list |
| Basis | A basis of a vector space V is a list of vectors in V that is linearly independent and spans V. Any two bases have the same length |
| Dimension | The dimension of a finite-dimensional vector space V (denoted by dim(V)) is the length of any basis of the vector space |
| Spanning list reduction | Every spanning list in a vector space can be reduced to a basis of the vector space |
| Existence of bases | Every finite-dimensional vector space has a basis |
| Linearly independent list expansion | Every linearly independent list of vectors in a finite-dimensional vector space can be extended to a basis of the vector space |
| Linearly independent / spanning list length | Every linearly independent list of vectors in V of length dim(V) is a basis of V. Every spanning list of vectors in V of length dim(V) is a basis of V. |
| If A is a subspace of a finite dimensional vector space V | Then A is also finite dimensional |
| If v1, v2 are subspaces of a finite dimensional vector space V | Then dim(v1+v2) = dim(v1) + dim(v2) - dim(v1∩v2) |
| If v1, v2 are subspaces of a finite dimensional vector space V and form a direct sum | Then dim(v1+v2) = dim(v1) + dim(v2) |
| A list of vectors v1,...,vn in a finite dimensional vector space V is a basis of V iff | Every u in V can be written uniquely in the form u = a1v1+...+anvn with ai in F |
| Suppose V is a finite dimensional vector space and U is a subspace of V | Then there exists a subspace W of V such that the direct sum of U and W is equal to V |
Created by:
mejones
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