linear algebra for dummies cheat sheet - for dummies.pdf

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Linear Algebra For DummiesFrom Linear Algebra For Dummies by Mary Jane SterlingTo study and solve linear algebra equations successfully, you need to know common numerical values oftrig functions, what elements determine a vector space, basic algebraic properties, and general commandsusing graphing calculators to solve linear algebra problems.

Commonly Used Values of Selected Trig FunctionsWhen performing transformations in trig functions, such as rotations, youneed to use the numerical values of these functions. Here are some of themore commonly used angles.

How to Meet Vector Space RequirementsIn linear algebra, a set of elements is termed a vector space when particularrequirements 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 thefollowing requirements:

Closure. u ⊕ v is in the set.

Commutativity. u ⊕ v = v ⊕ u.

Associativity. u ⊕ (v ⊕ w) = (u ⊕ v) ⊕ w.

Linear Algebra For Dummies Cheat Sheet - For Dummies http://www.dummies.com/how-to/content/linear-algebra-for-dummies-che...

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An identity element 0. u ⊕ 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 followingrequirements:

Closure. k ⊗ u is in the set.

Distribution over a vector sum. k ⊗ (u ⊕ v) = k ⊗ u ⊕ k⊗ v.

Distribution over a scalar sum. (k + l) ⊗ u = k ⊗u ⊕ l ⊗ u.

Associativity of a scalar product. k ⊗ (l ⊗ u) = (kl) ⊗ u.

Multiplication by the scalar identity. 1 ⊗ u = u.

Algebraic Properties You Should KnowYou can use a number of properties when working with linear algebraicexpressions, including the commutative, associative, and distributiveproperties of addition and multiplication, as well as identities and inverses inaddition and multiplication:

Calculator Commands for Linear AlgebraGraphing calculators are wonderful tools for helping you solve linear algebraprocesses; they allow you to drain battery power rather than brain power.Since there is a wide variety of graphing calculators out there, the followingare general instructions for help with linear algebra that apply to mostgraphing calculators:

To solve systems of equations by graphing:

1. Write each equation in y = mx + b form.1.

2. Insert equations in the y-menu.2.

3. Graph the lines.3.

4. Use the Intersection tool to get the answer.4.

To add or subtract matrices:

1. Insert the elements into the matrices A and B.1.


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2. With a new screen, press [A] + [B] or [A] – [B], and press Enter.2.

To multiply by a scalar:

1. Insert the elements into the matrix A.1.

2. With a new screen, press the scalar and multiply: k * [A], and pressEnter.

2.

To multiply two matrices together:

1. Insert the elements into the matrices A and B.1.

2. With a new screen, press [A] * [B], and press Enter.2.

To switch rows:

1. Insert the elements into a matrix.1.

2. Use row swap: rowSwap ([matrix name], first row, second row), andpress Enter.

2.

To add two rows together:


2. Use row addition: "row +", ([matrix name], row to be added to targetrow, target row), and press Enter.

2.

To add the multiple of one row to another:


2. Use row sum-of-multiple: "*row +", (multiplier, [matrix name], rowbeing multiplied, target row having multiple added to it), and pressEnter.

2.

To multiply a row by a scalar:


2. Use row multiple: "*row" (multiplier, [matrix name], row), and pressEnter.

2.

To create an echelon form:


2. Use row-echelon form: ref ([matrix name]) or reduced row-echelonform: rref ([matrix name]), and press Enter.

2.

To raise a matrix to a power:


2. Use the caret operation with power, p: [matrix name] ^ p, and pressEnter.

2.

To find inverses:


2. Use the reciprocal operation, x−1: [matrix name]−1, and press Enter.2.


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To solve systems of linear equations:(This only works when the system has a single solution; it fails when thematrix A is singular.)

1. Write each equation with the variables in the same order and theconstant on the other side of the equation sign.

1.

2. Create a matrix A, whose elements are the coefficients of thevariables.

2.

3. Create a matrix B, whose elements are the constants.3.

4. Press, A−1 * B, and press Enter.4.

The resulting vector has the values of the variables, in order.

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linear algebra for dummies cheat sheet - for dummies.pdf

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