Linea inearr Algeb lgebra ra For For Dummies Cheat She Sheet - For For Dummies ies
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Linear Algebra For For Dummies Dum mies From Linear Algebra For Dummies by Mary Jane Sterling To study and solve linear algebra equations successfully, you need to know common numerical values of trig functions, what elements determine a vector space, basic algebraic properties, and general commands using graphing calculators to solve linear algebra problems.
Commonly Used Values Values of Select Select ed Tri Tri g Functi ons When performing transformations in trig functions, such as rotations, you need to use the numerical values of these functions. Here are some of the more commonly used angles.
How t o Meet Vect Vect or Space Space Requirements In linear algebra, a set of elements is termed a vector space when particular requirements are met. For example, let a set consist of vectors u , v , and w . Also let k and l be real numbers, and consider the defined operations of ⊕ and ⊗. The set is a vector space if, under the operation of ⊕, it meets the following requirements: Closure. u
⊕
v is in the set.
Commutativity. u Ass ociat oc iat ivit iv it y. u
⊕
⊕
v=v
(v
⊕
⊕
u.
w ) = (u (u
⊕
v ) ⊕ w.
Linear Algebra For Dummies Cheat Sheet - For Dummies
An id ent it y el ement 0. u
⊕
http://www.dummies.com/how-to/content/linear-algebra-for-dummies-che...
0 = 0 ⊕ u = u for any element u.
An inverse element −u. u ⊕ −u
=
−u ⊕
u =0
Under the operation of ⊗, the set is a vector space if it meets the following requirements: Closure. k
⊗
u is in the set.
Distribution over a vector sum. k
(u
⊗
Distribution over a scalar sum. (k + l) Ass ociat ivit y o f a scal ar p rodu ct. k
⊗
⊕
v) = k
⊗
u =k
(l
⊗
⊗
u
⊕
k⊗ v.
l
⊗
u.
⊗
u.
⊗u ⊕
u) = ( kl)
Multipl ication by the scalar identity. 1 ⊗ u = u.
Al gebr aic Properti es You Shoul d Know You can use a number of properties when working with linear algebraic expressions, including the commutative, associative, and distributive properties of addition and multiplication, as well as identities and inverses in addition and multiplication:
Calcul ator Commands for Li near Al gebra Graphing calculators are wonderful tools for helping you solve linear algebra processes; they allow you to drain battery power rather than brain power. Since there is a wide variety of graphing calculators out there, the following are general instructions for help with linear algebra that apply to most graphing calculators:
To solve syst ems of equati ons by graphing: 1. 1. Writ e each equation in y = mx + b form. 2. 2. Insert equations i n th e y-menu. 3. 3. Graph t he lin es. 4. 4. Use the Intersection tool to get the answer.
To add or subtr act matri ces: 1. 1. Insert the elements into the matrices A and B.
Linear Algebra For Dummies Cheat Sheet - For Dummies
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2. 2. With a new screen, press [A] + [B] or [A] – [B], and press Enter.
To multi ply by a scalar: 1. 1. Insert the elements into the matrix A. 2. 2. With a new screen, press the scalar and multiply: k * [A], and press Enter.
To multiply two matrices together: 1. 1. Insert the elements into the matrices A and B. 2. 2. Wit h a new screen, press [A] * [B] , and press Enter.
To switch rows: 1. 1. Insert the elements into a matrix. 2. 2. Use row swap: rowSwap ([matrix name], first row, second row), and press Enter.
To add two rows together: 1. 1. Insert the elements into a matrix. 2. 2. Use row addition : " row +", ([matrix name], row to be added to target row, target row), and press Enter.
To add the multi ple of one row t o another: 1. 1. Insert the elements into a matrix. 2. 2. Use row sum -of -multiple: " *row +", (multipl ier, [matrix n ame], row being multi plied, target row h aving multipl e added to i t), and press Enter.
To multi ply a row by a scalar: 1. 1. Insert the elements into a matrix. 2. 2. Use row multiple: "*row" (multiplier, [matrix name], row), and p ress Enter.
To create an echelon for m: 1. 1. Insert the elements into a matrix. 2. 2. Use row -echelon form: ref ([matrix name]) or reduced row-echelon form: rref ([matrix name]), and p ress Enter.
To rais e a matr ix to a power: 1. 1. Insert the elements into a matrix. 2. 2. Use the caret operation with power, p: [matrix name] ^ p, and press Enter.
To fi nd i nvers es: 1. 1. Insert the elements into a matrix. 2. 2. Use the reciprocal operation, x −1: [matrix name] −1, and press Enter.
Linear Algebra For Dummies Cheat Sheet - For Dummies
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To solve systems of l inear equations: (This only works when the system has a single solution; it fails when the matrix A is singular.) 1. 1. Write each equation with the variables in the same order and the constant on the other side of the equation sig n. 2. 2. Create a matrix A, who se elements are the coeffici ents of t he variables. 3. 3. Create a matrix B, w hose elements are the cons tants. 4. 4. Press, A
−1
* B, and press Ent er.
The resulting vector has the values of the variables, in order.
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