Vector and Matrix Operations Quiz

Addition: Component-wise addition of two vectors
- Example: [a, b] + [c, d] = [a+c, b+d]
Scalar Multiplication: Multiplication of a vector by a number
- Example: 2 * [a, b] = [2a, 2b]
Dot Product (Inner Product): Scalar product of two vectors
- Example: [a, b] · [c, d] = ac + bd
Norm (Magnitude): Length of a vector
- Example: ||[a, b]|| = √(a^2 + b^2)

Addition: Component-wise addition of two matrices
- Example: [a, b; c, d] + [e, f; g, h] = [a+e, b+f; c+g, d+h]
Scalar Multiplication: Multiplication of a matrix by a number
- Example: 2 * [a, b; c, d] = [2a, 2b; 2c, 2d]
Matrix Multiplication: Product of two matrices
- Example: [a, b; c, d] * [e, f; g, h] = [ae+bg, af+bh; ce+dg, cf+dh]
Inverse: Matrix that, when multiplied, results in the identity matrix
- Example: A * A^-1 = I

Linear Transformation: Function that satisfies linearity properties
- Example: T(v) = Av, where A is a matrix
Image: Set of vectors resulting from applying the transformation
- Example: Im(T) = {T(v) | v is a vector}
Kernel: Set of vectors that map to the zero vector
- Example: Ker(T) = {v | T(v) = 0}

Eigenvalue: Scalar that satisfies the equation Ax = λx
- Example: λ is an eigenvalue if Ax = λx has a non-trivial solution
Eigenvector: Non-zero vector that, when transformed, results in a scaled version
- Example: x is an eigenvector if Ax = λx

Determinant: Scalar value that can be used to describe the linear transformation
- Example: det(A) = ad - bc for a 2x2 matrix A = [a, b; c, d]
Properties:
- det(AB) = det(A)det(B)
- det(A^-1) = 1/det(A)

Addition: The result of adding two vectors is a new vector with components that are the sum of the corresponding components of the original vectors.
Scalar Multiplication: Multiplying a vector by a number results in a new vector with components that are the product of the original components and the number.
Dot Product: The dot product of two vectors is a scalar value that is the sum of the products of the corresponding components of the two vectors.
Norm: The norm of a vector is its length, which can be calculated using the Pythagorean theorem.

Addition: Matrix addition is performed component-wise, resulting in a new matrix with components that are the sum of the corresponding components of the original matrices.
Scalar Multiplication: Multiplying a matrix by a number results in a new matrix with components that are the product of the original components and the number.
Matrix Multiplication: The product of two matrices is a new matrix with components that are the dot product of the rows of the first matrix and the columns of the second matrix.
Inverse: The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.

Linear Transformation: A linear transformation is a function that satisfies the linearity properties, meaning it preserves vector addition and scalar multiplication.
Image: The image of a linear transformation is the set of vectors that result from applying the transformation to all possible input vectors.
Kernel: The kernel of a linear transformation is the set of vectors that, when transformed, result in the zero vector.

Eigenvalue: An eigenvalue is a scalar that satisfies the equation Ax = λx, where A is a matrix and x is a non-zero vector.
Eigenvector: An eigenvector is a non-zero vector that, when transformed by a matrix, results in a scaled version of itself.

Determinant: The determinant of a matrix is a scalar value that can be used to describe the linear transformation represented by the matrix.
Properties: The determinant has several important properties, including det(AB) = det(A)det(B) and det(A^-1) = 1/det(A).

Consistent: A system of linear equations is consistent if it has at least one solution.
Inconsistent: A system of linear equations is inconsistent if it has no solution.
Homogeneous: A system of linear equations is homogeneous if all the constants are equal to zero.
Non-homogeneous: A system of linear equations is non-homogeneous if at least one constant is not equal to zero.

A vector is a mathematical object with both magnitude and direction, often represented graphically as an arrow in a coordinate system
A vector space is a set of vectors that can be added together and scaled (multiplied by a number), satisfying certain axioms

A norm is a function that assigns a non-negative value to each vector, used to measure its magnitude or length
Inner product (dot product): a way to combine two vectors, resulting in a scalar value, used to define the angle between vectors

Mathematical vectors can be represented as numerical arrays, enabling computational manipulation
Vector representations are used in various applications, such as computer graphics, machine learning, and physics
Vector data can be characterized by a mean vector and a covariance matrix, summarizing its central tendency and variability

High-dimensional vector spaces exhibit unique properties, such as the concentration of measure phenomenon and the curse of dimensionality

A matrix is a rectangular array of numbers, often used to represent linear transformations
Matrices can be viewed as linear maps between vector spaces, applying transformations to input vectors

Basic geometric transforms, such as rotations and reflections, can be implemented using matrices
Matrices can be composed to perform complex transformations, enabling various applications in computer graphics and more