2050 Final exam cheat sheet PDF

Preview

Summary

Linear Algebra (Math 2050) final exam cheat sheet, includes all equations you may/will need for the final exam!...

Description

Important Rules and Formulas Dot Product:

𝑎 𝑑 Given 𝑢󰇍 = [ 𝑏], 𝑣 = [ 𝑒 ]. 󰇍𝑢 ∙ 𝑣 = 𝑎𝑑 + 𝑏𝑒 + 𝑐𝑓 𝑓 𝑐

Properties of the Dot Product: 1. 𝑢󰇍 ∙ 𝑣 = 𝑣 ∙ 𝑢󰇍

󰇍󰇍 ) = 𝑢 󰇍 𝑣 + 𝑢󰇍 𝑤 󰇍󰇍 2. 𝑢󰇍 ∙ (𝑣 + 𝑤 (𝑢󰇍 + 𝑣) ∙ 𝑤 󰇍󰇍 = 𝑢 󰇍𝑤 󰇍󰇍 + 𝑣𝑤 󰇍󰇍

3. (𝑐𝑢󰇍 ) ∙ 𝑣 = 𝑐(𝑢󰇍 ∙ 𝑣) = 𝑢 󰇍 ∙ (𝑐𝑣) 4. ‖ 𝑢󰇍 ‖2= 𝑢 󰇍 ∙ 󰇍𝑢

3. (𝑐𝑢󰇍 ) × 𝑣 = 𝑐(𝑢󰇍 × 𝑣) = 𝑢󰇍 × (𝑐𝑣) 4. (𝑢 󰇍 × 𝑣) × 󰇍𝑤 󰇍 ≠𝑢 󰇍 × (𝑣 × 𝑤 󰇍󰇍 )

5. 𝑢󰇍 × 0 = 0 × 𝑢 󰇍

6. 𝑢󰇍 × 𝑣 is orthogonal to 𝑢󰇍 and 𝑣 7. 𝑢󰇍 × 𝑢󰇍 = 0

8. 𝑢󰇍 × 𝑣 = 0 if and only if 𝑢󰇍 and 𝑣 are parallel

between vectors)

󰇍𝑢󰇍 ∙𝑣󰇍 ‖𝑢 󰇍󰇍 ‖∙‖𝑣󰇍 ‖

1. cos 𝜃 =

2. 𝑢󰇍 ⊥ 𝑣 ⟺ 𝑢󰇍 ∙ 𝑣 = 0

𝑎 Given 𝑢󰇍 = [𝑏 ], then ‖ 𝑢 󰇍 ‖ = √𝑎 2 + 𝑏 2 + 𝑐 2 𝑐 Length

Cauchy-Bunyakovsky-Schwarz Inequality |𝑢󰇍 ∙ 𝑣| ≤ ‖ 𝑢󰇍 ‖ ∙ ‖ 𝑣‖ Geometric – Arithmetic Mean Inequality 2

× 𝑣= |

Area: A parallelogram = ‖ 𝑢󰇍 × 𝑣‖ (where 𝑢󰇍 and 𝑣 are adjacent sides)

Atriangle = 2 ‖ 𝑢 󰇍 × 𝑣‖ 1

Lagrange’s Identity: ‖ 𝑢󰇍 × 𝑣‖2 = ‖ 𝑢󰇍 ‖2 ∙ ‖ 𝑣‖2− (𝑢󰇍 ∙ 𝑣 )2 Projections onto Vectors and Lines: projv𝑢󰇍 =

𝑢󰇍 ∙𝑣󰇍 𝑣 󰇍 ∙𝑣 𝑣 󰇍

Projections onto Planes through the Origin:

𝑎+𝑏

󰇍 ∙𝑒 𝑤

proj π 𝑤 󰇍󰇍 = 𝑒∙𝑒 𝑒 +

Cross Product: 𝑑 𝑎 Given 𝑢󰇍 = [𝑏 ] and 𝑣 = [ 𝑒 ] 𝑓 𝑐 󰇍 󰇍𝑢

󰇍 ) = 󰇍𝑢 × 𝑣 + 𝑢󰇍 × 𝑤 󰇍 2. 𝑢󰇍 × (𝑣 + 𝑤

9. ‖ 𝑢󰇍 × 𝑣‖ = ‖ 𝑢󰇍 ‖ ∙ ‖ 𝑣‖ sin 𝜃 (where 𝜃 is angle

Angles

√𝑎𝑏 ≤

Properties of the Cross Product: 1. 𝑢󰇍 × 𝑣 = −𝑣 × 𝑢󰇍

𝑖𝑎 𝑗𝑏 𝑘𝑐 𝑑

𝑒 𝑓

|

𝑤󰇍󰇍 ∙𝑓 𝑓∙𝑓

𝑓

Idempotent Matrix:

Is a matrix 𝐴 where 𝐴2 = 𝐴 Commutable Matrices:

If A and B commute, then 𝐴𝐵 = 𝐵𝐴

𝑢 × 𝑣 = (𝑏𝑓 − 𝑐𝑒)𝑖 − (𝑎𝑓 − 𝑐𝑑)𝑗 + (𝑎𝑒 − 𝑏𝑑)𝑘

Properties of Matrix Addition and Scalar Multiplication: 1. If 𝐴 and 𝐵 are matrices, then 𝐴 + 𝐵 is a matrix. 2. 𝐴 + 𝐵 = 𝐵 + 𝐴

3. (𝐴 + 𝐵 ) + 𝐶 = 𝐴 + (𝐵 + 𝐶)

4. 𝐴 + 0 = 0 + 𝐴 = 𝐴 , where zero is a matrix. 5. 𝐴 + (−𝐴) = (−𝐴) + 𝐴 = 0

6. c𝐴 is a matrix with same dimensions as 𝐴. 7. 𝑐(𝑑𝐴) = (𝑐𝑑)𝐴 8. 1𝐴 = 𝐴

10. If 𝑐𝐴 = 0, then either 𝑐 = 0 or 𝐴 = 0. Properties of Matrix Multiplication 1. (𝐴𝐵)𝐶 = 𝐴(𝐵𝐶 )

2. 𝑐(𝐴𝐵) = (𝑐𝐴)𝐵 = 𝐴(𝑐𝐵)

3. (𝐴 + 𝐵 )𝐶 = 𝐴𝐶 + 𝐵𝐶 and 𝐴(𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶 4. 𝐴0 = 0 and 0𝐴 = 0 , where 0 is the zero matrix.

5. 𝐼𝑛 is a 𝑛 × 𝑛 identity matrix and behaves like 1. 𝐴𝐼𝑛 = 𝐴 and 𝐼𝑛 𝐵 = 𝐵

6. 𝐴𝐵 ≠ 𝐵𝐴 or 𝐵 = 0.

Properties of the Transpose of a Matrix 1. (𝐴 + 𝐵)𝑇 = 𝐴𝑇 + 𝐵𝑇

2. (𝑐𝐴) = 𝑐𝐴

3. (𝐴 ) = 𝐴

𝑇

5. (𝐴𝑇 )−1 = (𝐴−1 )𝑇

If 𝐴 = [𝑎 𝑏 ], then 𝐴−1 = 𝑐 𝑑 𝑎𝑑 − 𝑏𝑐 ≠ 0

3. (𝐴𝐵)−1 = 𝐵−1 𝐴−1

4. (𝑘𝐴)−1 = 𝑘 −1 𝐴−1 , for non-zero scalar 𝑘 5. (𝐴𝑇 )−1 = (𝐴−1 )𝑇

6. A matrix can NOT have more than one inverse matrix |𝑎 𝑏 | = 𝑎𝑑 − 𝑏𝑐 𝑐 𝑑

Properties of Determinants 1. The determinant of a triangular matrix is the product of its diagonal entries. 2. The determinant is linear in each row.

3. If a matrix has a row of zeroes, det = 0. 4. If a matrix has two equal rows, det = 0.

5. If one row of a matrix is a scalar multiple of another row, det = 0.

determinant changes sign.

7. det (𝑐 ∙ 𝐴) = 𝑐 𝑛 ∙ det 𝐴

8. (det 𝐴 ∙ 𝐵) = det 𝐴 ∙ det 𝐵

9. If A is invertible, then det 𝐴 ≠ 0 and det 𝐴−1 = det 𝐴 1

10. det 𝐴𝑇 = det 𝐴

11. det 𝐴𝑛 = (det 𝐴)𝑛

4. (𝐴𝐵)𝑇 = 𝐵 𝑇 𝐴𝑇

Inverse of a 2x2 Matrix

2. (𝐴−1 )−1 = 𝐴

6. If two rows of a matrix are interchanged, the

7. If 𝐴𝐵 = 0, it does not generally imply that 𝐴 = 0

𝑇 𝑇

1. If 𝐴−1 exists, then 𝐴𝐴−1 = 𝐼 and 𝐴−1 𝐴 = 𝐼

Determinant 2x2 Matrix

9. 𝑐(𝐴 + 𝐵) = 𝑐𝐴 + 𝑐𝐵 and (𝑐 + 𝑑 )𝐴 = 𝑐𝐴 + 𝑑𝐴

𝑇

Properties of Invertible Matrices

Using elementary row operations to find determinants:

1 𝑎𝑑−𝑏𝑐

[ 𝑑 −𝑏 ]; −𝑐 𝑎

Row Operation

Effect on determinant

Interchanging rows

(−1) det

𝑅𝑜𝑤𝑎 = 𝑅𝑜𝑤𝑎 ± 𝑐 ∙ 𝑅𝑜𝑤𝑏

𝑐 det

no effect

Multiplying a row by scalar 𝑐

Inverse of a 3x3 Matrix Similarity & Diagonalizability Finding Inverse using Adjoint and Determinant If 𝐴−1 exists, then [ 𝐴 | 𝐼3 ] → [ 𝐼3 | 𝐴−1 ]

Characteristic Polynomial: 𝑃(𝜆) = det (𝐴 − 𝜆𝐼) −1

1 𝐴−1 = det 𝐴 adj 𝐴

Matrices 𝐴 and 𝐵 are similar if: 𝑃 𝐴𝑃 = 𝐵 diagonal matrix. if: 𝑃 −1 𝐴𝑃 = 𝐷 , where D is a A is diagonalizable

Three points

Line thru 2 points

Plane thru 3 points

Intersection

Distance

Distance

Distance

Distance

Distance

Linear combination...