Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 ↓


Vectors and Matrices


Vectors and matrices are fundamental concepts in mathematics, especially in the field of linear algebra. They are important for solving systems of equations, performing transformations, and representing various mathematical models. Let's dive into the details of vectors and matrices by understanding their definitions, operations, and applications.

Understanding vector

A vector can be thought of as a list of numbers that describes a quantity with both magnitude and direction. Vectors are often used to represent physical quantities such as force, velocity, and acceleration. A simple visual example of a vector is an arrow pointing from one position in space to another.

Notation and representation

Vectors are usually represented using letters such as v or u with an arrow at the top: (vec{v}) or (vec{u}). However, in plain text, boldface is often used, such as v or u.

In two-dimensional space, a vector can be represented as:

V = [x, y]

In three-dimensional space, a vector takes the following form:

V = [x, y, z]

Here, x, y and z are the components of the vector, indicating its position along the X, Y and Z axes, respectively.

Vector addition

Vector addition is simple and follows the "head-to-tail" rule. Consider two vectors:

A = [A1, A2]
b = [b1, b2]

Their sum c is calculated by adding the corresponding components:

c = a + b = [(a1 + b1), (a2 + b2)]

Visual:

A B A+B

Scalar multiplication

Vectors can be multiplied by a scalar (a real number) to increase or decrease their magnitude. If k is a scalar and v = [v1, v2] is a vector, then scalar multiplication is as follows:

k * v = [k * v1, k * v2]

If k = 2 and v = [1, 3], then:

2 * [1, 3] = [2, 6]

Understanding matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. They are used to solve systems of linear equations, perform transformations, and manage data structures in various fields such as computer graphics.

Matrix notation

The matrix is usually denoted by a capital letter, such as A, and looks like this:

A = [A11 A12]
    [A21 A22]

This is a 2x2 matrix, where a11, a12, a21 and a22 are the elements of the matrix.

Matrix dimensions

The dimensions of a matrix are given in the form mxn, where m is the number of rows and n is the number of columns. For example, a matrix:

b = [1 2 3]
    [4 5 6]
    [7 8 9]

Its dimension is 3x3.

Matrix addition

Matrix addition is similar to vector addition and is possible only when both the matrices have the same dimensions.

Consider two matrices:

C = [C11 C12]
    [C21 C22]

D = [D11 D12]
    [D21 D22]

Their sum is calculated elementwise:

C + D = [C11 + D11, C12 + D12]
        [C21 + D21, C22 + D22]

Scalar multiplication of matrices

Just like vectors, matrices can also be multiplied by scalars.

If k is a scalar and E is a matrix:

E = [E11 E12]
    [E21 E22]

The scalar multiplication is:

k * e = [k * e11, k * e12]
        [K*E21, K*E22]

Matrix multiplication

Matrix multiplication is a bit more complicated and is only defined when the number of columns in the first matrix matches the number of rows in the second matrix.

Consider two matrices:

f = [f11 f12]
    [F21 F22]

G = [G11 G12]
    [G21 G22]

The product of F and G is:

f * g = [(f11 * g11 + f12 * g21) (f11 * g12 + f12 * g22)]
        [(f21 * g11 + f22 * g21) (f21 * g12 + f22 * g22)]

Identity matrix

The identity matrix is a square matrix with 1's on the diagonal and 0's elsewhere. It is the matrix equivalent to the number 1.

For example, the 3x3 identity matrix:

I = [1 0 0]
    [0 1 0]
    [0 0 1]

Any matrix multiplied by the identity matrix remains unchanged.

Determinants and inverses

The determinant is a special number that can be calculated from a square matrix. It provides information about the matrix such as whether it has an inverse or not.

For a 2x2 matrix:

J = [J11 J12]
    [J21 J22]

The determinant is calculated as follows:

det(J) = j11 * j22 - j12 * j21

If the determinant is not zero, then the matrix has an inverse, denoted by J -1, and is calculated as:

J -1 = (1/dit(J)) * [J22 -J12]
                              [-j21 j11]

Applications of vectors and matrices

Vectors and matrices are used in various applications in different fields.

Physics and engineering

Vectors help represent quantities such as force and velocity. Matrices help analyze stress and strain in materials.

Computer graphics

Matrices are used for transformations such as rotation, scaling, and translation of images and objects.

Economics

Matrices help in modeling economic systems and solving optimization problems.

Computer science

In computer science, matrices are fundamental in algorithms, data structures, and networks.

Figures

Matrices are crucial for multidisciplinary data analysis.

Conclusion

Understanding vectors and matrices provides a foundation for further study in mathematics and its applications in various sciences. Mastering these topics is essential for engineers, scientists, and anyone working with complex systems.


Grade 11 → 5


U
username
0%
completed in Grade 11

← Prev (4.5.3)
Average Value of a Function
Next (5.1) →
Vectors

Comments 



Vectors and Matrices

https://www.buddymath.com/g11/5?bmal=en