Matrix Algebra and Differential Equations Essentials
Matrix Rank
- Echelon Form: rank = number of non-zero rows
- Normal Form: reduce to identity matrix (I); rank = number of 1s on diagonal
Matrix Forms
- Echelon Form: zeros below pivots
- Normal Form (RREF): pivots = 1 and only non-zero in column
Systems of Equations
- Form augmented matrix [A|B], row reduce
- Consistent if no row like [0 0 0 | b≠0]
- Unique: rank A = rank [A|B] = number of variables
- Infinite: rank A = rank [A|B] < number of variables
- No solution: rank A ≠ rank [A|B]
Linear Dependence/Independence
- Vectors
HDLC and PPP Data Link Protocols Explained
HDLC Frame Control Fields
The Information (I) field is used to transport user data and network layer data. It includes information for error and flow control (where piggybacking is applicable).
It contains a 1-bit Poll/Final (P/F) bit (if it’s 0, it’s an I-frame) and the next 3 bits: N(S) (send sequence number) and the last 3 bits: N(R) (receive sequence number), which is the acknowledgment field when piggybacking is used.
The N(S) and N(R) bits are only meaningful when the P/F bit is active. The P/
Read MoreCalculus Proofs: Continuity, Derivatives, and Limits
1. Let f and g be continuous functions from D to R. Prove that f + g is continuous on D using the definition of continuity.
Proof: We must show that f + g is continuous at c for each c ∈ D. Let c ∈ D and ε > 0. Since f is continuous at c, there exists a δ₁ > 0 such that if x ∈ D and |x – c| < δ₁, then |f(x) – f(c)| < ε/2. By the continuity of g at c, there exists a δ₂ > 0 such that if x ∈ D and |x – c| < δ₂, then |g(x) – g(c)| < ε/2. Let δ = min{δ₁,
Read MoreBusiness Statistics and Data Analysis for Managerial Decisions
Statistics and Business Analytics: Definitions, Needs, and Importance
Statistics is the science of collecting, organizing, analyzing, interpreting, and presenting data. It helps in converting raw data into meaningful information for decision-making.
Business Analytics refers to the use of statistical methods, data analysis, predictive modeling, and fact-based management to drive business planning. It focuses on turning data into actionable insights to solve business problems and improve performance.
Read MoreDiscrete Mathematics & Probability Essentials
Probability Fundamentals
Probability is defined as the ratio of the number of successful outcomes to the total number of possible outcomes. Every probability problem involves a probability space, which consists of:
- Sample Space: A non-empty, countable set containing all possible outcomes of an experiment.
- Probability Function: A function that maps each outcome to its probability. The sum of probabilities for all outcomes in the sample space must equal 1.
Core Probability Rules
Complement Rule
P(not A)
Read MoreGeometry Problems: Circles, Chords, and Quadrilaterals
Geometry Problems and Solutions
Here are the solutions to the geometry questions:
Cyclic Quadrilateral Angle Sum
Question: The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. If ‘n’ represents this sum and ‘m’ is also 180°, find ‘m-n’.
Solution: In a cyclic quadrilateral, the sum of opposite angles is always 180°. Therefore, \(n = 180^\circ\). Given \(m = 180^\circ\), we have:
\[m – n = 180^\circ – 180^\circ = 0\]
Circle’s Largest Chord and Radius
Question: If the largest chord