A Deep Dive into Subspaces, Vector Spaces, and Their Applications in Linear Algebra
What is a Subspace of R^n?
A set U of vectors in R^n is called a subspace of R^n if it satisfies the following properties:
- The zero vector 0-> ∈ U.
- If x-> ∈ U and y-> ∈ U, then x->+y-> ∈ U.
- If x-> ∈ U, then ax-> ∈ U for every real number a.
Any subspace of R^n other than {0} or R^n is called a proper subspace.
Why is a Plane Through the Origin a Subspace of R^3?
Every plane M through the origin in R^3 has equation ax + by + cz = 0 where a, b, and c are not all zero. Here
n-
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Determinants and Invertibility
What is the Determinant of a Matrix?
The determinant of a matrix is a scalar value that provides information about the matrix’s invertibility. Only square matrices have determinants.
Calculating Determinants
For a 2×2 matrix A = [a b; c d], the determinant is calculated as det(A) = ad – bc.
For 3×3 matrices and larger, methods like cofactor expansion or row reduction are used.
Determinants and Row Operations
Elementary row operations affect the determinant as follows:
- Interchanging
Understanding Asset Management: A Comprehensive Guide
Defining Asset Management
Administration, derived from the Latin terms ad-minister (“to serve”) and ad manus trahere (“to manage”), is crucial for the effective functioning of organizations. It involves the efficient management of both tangible and intangible resources to achieve desired goals. People, money, technology, and even time can be considered resources depending on the context.
The Importance of Resource Management
Effective resource allocation ensures optimal operational efficiency. Various
Read MoreStatistics Cheat Sheet: Key Concepts and Formulas
Variables and Confounding
Types of Variables
Numerical (Measurable):
- Continuous (Range): Values can fall within a range.
- Discrete (Limited Value): Values have specific, limited possibilities.
Categorical:
- Nominal (Unordered): Categories have no inherent order.
- Ordinal (Ordered): Categories have a natural order.
Confounding Variables
A confounding variable is associated with both the explanatory and response variables, potentially influencing the observed relationship between them. It is not a consequence
Read MoreAccounting Concepts and Transactions
Bank’s Cash Balance: Company’s Cash Balance: – Gross profit = net sales revenue – cost of goods sold
+ Deposits outstanding + Notes received by bank – Initial Inventory + Purchased Inventory – Ending Inventory = COGS
– Checks outstanding + Interest received
– Unrecorded payments
Read MoreFinance Exam Study Guide: Key Concepts & Formulas
Final Exam Study Guide
Chapter 4: Time Value of Money
Present Value (PV)
The current value of future cash flows discounted at the appropriate discount rate, calculated at t = 0 on a timeline.
Formula: PV = FV / (1+r)^t
Discount Rate
The rate of return used to determine the present value of future cash flows. It represents the minimum rate of return required by an investor or business to undertake an investment project or value an investment opportunity.
Future Value (FV)
The amount an investment is worth
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