# Shared Flashcard Set

## Details

Linear Algebra 1st Midterm
Linear Equations
16
Mathematics
02/19/2012

## Additional Mathematics Flashcards

Term
 Matrix
Definition
 Rectangular array of numbers
Term
 mxn matrix
Definition
 m = rows n = columns
Term
 Elementary Row Operations
Definition
 1. Multiply a row by constant 2. Add a multiple of a row to another and replace row 3. Swap or interchange rows
Term
 Row Echelon Form
Definition
 1. All non zero rows are above zero rows 2. Each leading non zero entry of a row is in a column to the right of the one above it 3. All entries in a column beneath a leading column must be 0
Term
 Reduced Row Echelon Form
Definition
 1. Each leading entry is equal to 1 2. Each lead entry is the only non zero entry in its column
Term
 Free Variable
Definition
 A variable x in a system of linear equations in n unknowns x1 . . . xn is free if there are no restrictions placed on that variable
Term
 Uniqueness Theorem
Definition
 A linear system is consistant <==> the rightmost column of the augmented matrix is NOT a pivot column
Term
 Span
Definition
 V is an element of the span {v1,...vn} if v is a linear combination of v1,...,vn  Span {v1...vn} is the set of all linear combinations of v1...vn
Term
 Matrix Equation
Definition
 Ax=b where A is a matrix and x,b are vectors
Term
 Linear Dependence
Definition
 A set of vectors {v1...vn} is said to be linearly dependent if there exists scalars x1...xn, not all of them zero, such that x1v1 + . . .+ xnvn = 0 (ld relation)  So at least one of the vectors can be written as a lc of the others
Term
 Linear Independence
Definition
 A set {v1...vn} are linearly independent if x1v1 + xnvn= 0 has only the trivial solution x1 = xn = -0
Term
 Tests for linear dependence
Definition
 1. Set contains 0 vector 2. v1 and v2 arent 0 vector and v1 = kv2 (scalar multiples) 3. A set contains more vectors than entries
Term
 Transformations
Definition
 A transformation T from Rn to Rm is a rule that assigns to each vector x in Rn a vector T(x) in Rm
Term
 Domain of T Codomain of T Range of T
Definition
 Rn domain Rm range Set of all images t(x)
Term
 A transformation is linear if:
Definition
 1. T(u + v) = T(u) + T(v) for all u,v in the domain of T 2. T(cu) = cT(u) for all u and scalars c
Term
 If T is linear then:
Definition
 T(0) = 0 vector   T(cu + dv) = cT(u) + dT(v)
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