Linear algebra Flashcards
What are vectors
Vectors are mathematical objects used to represent quantities that have both magnitude and direction.
How are vectors represented
Represented by an ordered collection of numbers, typically arranged in a column or row format.
What are properties of vector
Magnitude
Direction
Addition
Scalar Multiplication
Subtraction
Zero Vector
Negative Vector
What is magnitude of a vector
Every vector has a magnitude, which represents the length or size of the vector. The magnitude of a vector is always a non-negative real number.
What is direction of a vector
Vectors have a direction, indicating the orientation or angle of the vector in space.
Addition of two vectors
Vectors can be added together using the parallelogram rule or the head-to-tail method. The sum of two vectors is another vector obtained by connecting the initial point of the first vector to the terminal point of the second vector.
Scalar multiplication of a vector
Vectors can be multiplied by scalars (real numbers). Scalar multiplication changes the magnitude of the vector without altering its direction.
Subtraction of vectors
Vector subtraction is equivalent to adding the negative of a vector. Subtracting a vector is the same as adding its negative.
What is zero vector
There exists a unique vector called the zero vector, denoted as
[0]
[0] , which has zero magnitude and undefined direction. Adding the zero vector to any vector results in the original vector.
What is negative vector
Every vector has a negative counterpart, obtained by reversing its direction while keeping its magnitude unchanged.
Formula for calculating magnitude
∣∣v∣|2= sqrt(v12+v22+…+vn**2)
Formula for calculating direction of a vector
Different vector operations
Vector Addition
Scalar Multiplication
Vector Subtraction
Dot Product
Cross Product
Dot product is also called as
Scalar product
Cross product is also called as
Vector product
What is dot product
The dot product between two vectors is based on the projection of one vector onto another. Let’s imagine we have two vectors a
and b, and we want to calculate how much of a
is pointing in the same direction as the vector b
. We want a quantity that would be positive if the two vectors are pointing in similar directions, zero if they are perpendicular, and negative if the two vectors are pointing in nearly opposite directions. We will define the dot product between the vectors to capture these quantities.
Another name for dot product or scalar product
Inner product
Formula for dot product
v⋅u=v1u1 + v2 u2 + …+ vnun
What does dot product geometrically represent
Geometrically, the dot product represents the projection of one vector onto another, scaled by the length of the other vector.
What is the norm of a vector
The norm of a vector, also known as its magnitude or length, represents the distance of the vector from the origin in the vector space.
Norm of a vector is vector or scalar?
Scalar
Euclidean Norm (L2 Norm):
||v||₂ = √(v₁² + v₂² + … + vₙ²)
Taxicab Norm (L1 Norm):
||v||₁ = |v₁| + |v₂| + … + |vₙ|
Maximum Norm (L∞ Norm):
||v||₊ = max(|v₁|, |v₂|, …, |vₙ|)
What is cross product?
The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. Unlike the dot product, which results in a scalar quantity, the cross product yields a vector that is perpendicular to both input vectors.
cross product formula
v × u = [v₂u₃ - v₃u₂, v₃u₁ - v₁u₃, v₁u₂ - v₂u₁]
What does cross product represent geometrically?
Geometrically, the magnitude of the vector from cross product represents the area of the parallelogram formed by the two input vectors, and its direction is perpendicular to this parallelogram. The right-hand rule is often used to determine the direction of the resulting vector.
What is the angle of a vector
The angle between two vectors can be determined using the dot product formula and trigonometric functions. Given two vectors v and u, the angle θ between them can be found using the formula:
θ=arccos(v⋅u/∣∣v∣∣×∣∣u∣∣ )
θ=arccos(v⋅u/∣∣v∣∣×∣∣u∣∣ ) - this formula gives the angle in
Radians
||v|| and ||u|| are
The magnitudes (norms) of the vectors.
Magnitude of a vector or
Eucledian distance of a vector
Eucledian distance is denoted by
||v|| subscript 2 - since using Eucledian norm L2 formula or |v|
Projection of a vector
The projection of a vector onto another vector is the component of the first vector that lies in the direction of the second vector. It represents how much of one vector acts in the direction of another vector.
Formula for projection of a vector
proj-sub-u (v)=(v⋅u/||u||**2)u
What is linear independence
A set of vectors in a vector space such that none of the vectors can be expressed as a linear combination of the others. In simpler terms, a set of vectors is linearly independent if no vector in the set can be formed by multiplying another vector by a scalar and adding it to the others.
What is the dot product of orthogonal vectors
0
What is orthogonality of vectors
Orthogonality of vectors refers to the property where two vectors are perpendicular to each other in a vector space. Geometrically, this means that the angle between the vectors is 90 degrees (or π/2 radians).
What is matrix
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.
What is matrix determinant?
The determinant of a square matrix is a scalar value that represents certain properties of the matrix.
What is linear transformation
It is a mapping that takes vectors from one vector space to another in such a way that straight lines remain straight and parallel lines remain parallel after transformation.
How is determinant represented
det(A) or |A|
Formula for det
det(A)= ad-bc
A matrix is invertible if and only if its det is
non zero
For a matrix that has linearly dependent columns, det is
zero
det(A) = 0 states that
- Matrix A is not invertible
- Vectors of the matrix lie in the same sub space/dependent
What is the rank of a matrix
The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
Formula for rank -
rank(A)=max number of linearly independent rows or columns
Rank of a matrix can be calculated by
- row reduction
- calculating det of sub matrices
- SVD
What is a singular matrix?
A square matrix that is not invertible.
Determinant is only calculated for
Square matrices
A singular matrix is also called as
degenerate matrix
What does it mean a matrix is not invertible
Its determinant is zero
Rank properties when a square matrix A is invertible
If a square matrix A is invertible (non-singular), its rank is indeed equal to the number of rows (or columns), but it’s more precise to say that its rank is equal to its dimension. (square mxn, m=n - dimension m)
Rank properties when a square matrix A is singular
If a square matrix. A is singular, its rank is less than n.
Singular matrices have several important properties:
- They do not have full rank, meaning they do not have enough linearly independent rows or columns to span the entire space.
- They cannot be inverted. Attempting to compute the inverse of a singular matrix will result in an error or an incorrect result.
- They represent transformations that collapse or distort space, leading to loss of information.
What is an invertible matrix?
An invertible matrix, also known as a non-singular matrix or non-degenerate matrix, is a square matrix that has an inverse. In other words, it’s a matrix that can be multiplied by another matrix to produce the identity matrix, and vice versa.
For nxn square matrix A, there exists a square matrix B(inverse of A) such that A . B = B.A = I
Inverse is denoted by
A**-1
Transpose of a matrix
The transpose of a matrix is an operation that flips the matrix over its diagonal, switching its rows and columns. A**T
Transposing a matrix, changes the rank. True or False
False. Transposing a matrix doesn’t change the rank
Properties of matrix transpose
Double Transpose: (A’’) = A
Transpose of a Sum: (A + B)’ = A’ + B’
Transpose of a Scalar Multiple: (kA)’ = k(A’)
Transpose of a Product: (AB)’ = B’A’
Transpose of an Inverse: (A^(-1))’ = (A’)^(-1)
Eigenvalues and eigenvectors are concepts in linear algebra that are associated with_____ matrices.
square
Eigenvalues
Eigenvalues are scalars that represent the scaling factor of eigenvectors when a linear transformation is applied.
Av=λv
Eigenvectors
Eigenvectors are non-zero vectors that are transformed only by a scalar factor (the eigenvalue) when a linear transformation is applied.
Properties of eigen vectors/values
- Eigen vector of a matrix when multiplied doesnt change the direction of the vector. It is just scaled via the eigen value
- Eigenvectors corresponding to different eigenvalues are linearly independent.
- Eigen values of skew-symmetric matrices are either imaginary or zero
- The sum of eigen values of a matrix is equal to the trace.
- If A is a square matrix, eigen value= 0 is not eigen value
- The scalar multiple of eigen value is also an eigen value of A
- Product of eigen values is equal to the determinant of A
Formula for calculating eigen values
- Det(A-lambdaI)
- Find descriminant
- lambda1 = -b - sqrt(Discrim)/2a
- lambda2 = -b + sqrt(Discrim)/2a
NXN matrix has __ eigen values/eigen vectors
N eigen values
Calculate eigen vectors from eigen values
What is unit vector
How to calculate unit vector
Scalar projection vs vector projection
a scalar projection is the length of a vector projected onto another vector. A vector projection is the vector length plus the direction onto another vector.
