GRE Study Linear Algebra

Description

Flashcards on GRE Study Linear Algebra, created by Marissa Miller on 05/10/2015.
Marissa Miller
Flashcards by Marissa Miller, updated more than 1 year ago
Marissa Miller
Created by Marissa Miller over 8 years ago
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Resource summary

Question Answer
Number of Solutions to a Linear System 0 solutions (inconsistent) 1 solution (consistent) infinitely many solutions (consistent)
Dot Product \( \vec{u}\cdot\vec{v} = u_{1}v_{1} + u_{2}v_{2} + \ldots + u_{n}v_{n}\)
Socks and Shoes Theorem \((AB)^{-1} = A^{-1}B^{-1} \)
Number of free variables = number of unknowns - number of nonzero rows in echelon matrix
Solution to \(A\vec{x} = \vec{b} \) if \(A\) is invertible \(\vec{x} = A^{-1}\vec{b} \)
Gaussian Elimination 1) augmented matrix 2) reduce to echelon form 3) backwards sub
Reducing to Echelon form options > multiply row by constant > interchange two rows > add multiple of another row
Inverse of a 2 by 2 Matrix \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] \[ A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
Vector Space > closed under addition and scalar multiplication > must contain \(\vec{0} \)
Nullspace > set of all solutions to \(A\vec{x} = \vec{0} \) > if a matrix is invertible, the only solution to \(A\vec{x} = \vec{0} \) is the trivial one
Linear Combination \(k_{1}\vec{v_{1}} + k_{2}\vec{v_{2}}+ \ldots k_{n}\vec{v_{n}}\)
Span set of all linear combinations of vectors
Linearly Independent If \(k_{1}\vec{v_{1}} + k_{2}\vec{v_{2}}+ \ldots k_{n}\vec{v_{n}} = \vec{0}\) is only true for \(k_{i} = 0\)
Basis collection of linearly independent vectors that span a space
Dimension number of vectors in a space
To determine of vectors are linearly independent... ...echelon the vectors; any free variables means they are dependent
Cross Product \(\vec{u}\times\vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \end{vmatrix} \)
Equation for plane through two vectors > cross product > use cross product to get a normal vector \( (a b c) \) > plane: \(ax +by +cz = 0\)
Column Rank max number of linearly independent columns
Row Rank max number of linearly independent rows
Rank = Row rank = column rank
Column Space for an m by n matrix, CS is a subspace of \( R^{m}\) spanning the columns
dim(CS(A)) = rank(A)
Basis for CS > echelon form of \(A^{T}\) > find number of independent columns > pick those columns
To see if \(\vec{b}\) is in CS(A)... ...see if there is a solution to \(A\vec{x} = \vec{b}\)
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