Vector Spaces (Linear Algebra)

Field

A set F of numbers with the property that if a and b are elements of F, then a+b, a-b, ab, and a/b are also all still in F. The set of natural numbers N, is not a field because N is the set of all positive integers, so if we took the number 3 from the set and the number 10 from the set, and applied the 4 operations above, we would have to have a corresponding value that is still in the set N. These numbers work for a+b, and ab because they give us 3+10=13, 13 is in the set of N (meaning it is a positive integer), and 3*10=30, 30 is in the set of N. However, if we did a-b, we would get 3-10=-7 which is not in the set N, -7 is an integer, but not positive. For a/b, we get 3/10, which is positive, but not an integer, so it is not contained in the set of N (all positive integers). Note:(elements of F are called scalars)

Basis

A set of vectors form a basis if these two conditions are met: 1. The set of vectors are linearly independent 2. The set of vectors span the given vector space. So they are the minimal spanning set of a vector space

Linear Dependence

A set of vectors is said to be linearly dependent, if there exists a non-trivial solution to the linear combination of vectors {v1,v2,v3....v_n} that equal the zero vector. If the matrix with these vectors as columns has a determinant of 0, it is linearly dependent. Another way to word it, if one of the vectors can be written as a linear combination of another vector in the set, then the set is linearly dependent, so if you can take a vector out, and it doesn't effect the span, then it is linearly dependent. So if one vector is a scaled version of another vector, the scaled vector is linearly dependent.

Linear Independence

A set of vectors is said to be linearly independent, if the solution to the system of equations is only the zero vector (trivial solution). For homogeneous systems, this happens when the determinant is non-zero.

Minimal Spanning Set

A span that has the minimum number of vectors needed to span a vector space.

Vector Space -bad definition, look at the one below

A vector space consists of a set vectors called V (elements of V are vectors), a field F (defined above), and two operations (scalar multiplication and vector addition). A vector space must also satisfy the list of axioms in the book, I am not writing them out.

Additive Inverse

A vector that when added to another vector, gives the zero vector. Example: for [x,y,z], the additive inverse would be [-x,-y,-z] because when you add the two you'd get [0,0,0].

Vector Addition

Adding each corresponding component of 2 different vector.

Helpful analogy for vector spaces

An easier way to think of a vector space is to think of it as a recipe. For a recipe you need: 1. A set I (where the elements of the set are ingredients) 2. Instructions on how much of each ingredient you need (scalar multiplication), and what to do with the ingredients to turn it into the final dish (vector addition).

Identity Matrix

A matrix that has 1's along the main diagonal, and 0's everywhere else. It doesn't transform a vector. The inverse of a matrix, multiplied by that original matrix, will give you an identity matrix. A*A^-1 = Identity matrix

Zero Vector

Behaves very similarly with other vectors, as the number 0 does with other numbers (Think of the properties of 0). The zero vector is a vector with 0's for every coordinate. Note however, there only exists 1 zero vector for any given number of dimensions. For example, the zero vector of 2 dimensions is (0,0), for 3 dimensions it is (0,0,0), and for n dimensions, the zero vector has n zero's for n coordinates.

TRUE OR FALSE: The vector (x, y) in R2 is the same as the vector (x, y, 0) in R3

FALSE

TRUE OR FALSE: The set {0, 1}, with the usual operations of addition and scalar multiplication, forms a vector space.

FALSE: 1 is not a vector, and there is no scalar field defined.

TRUE OR FALSE: The set Z of integers, together with the usual operations of addition and scalar multiplication, forms a vector space.

FALSE: A vector space is a set of vectors V over a a scalar field F granted it abides by the list of axioms. Z is not even a scalar field, much less a vector space.

TRUE OR FALSE: A linearly dependent spanning set, also be a minimal spanning set?

FALSE: If a set is linearly dependent, you can write one of the vectors as a linear combination of some other vectors in the vector space, so that one vector is redundant and extra since it doesn't tell us anything that the other vectors (that linearly combined to make the redundant vector) don't already. (couldn't think of another way to word it)

TRUE OR FALSE: If x is a vector in the first quadrant of R2, then any scalar multiple kx of x is still a vector in the first quadrant of R2

FALSE: If k is negative it will make the vector be in the third quadrant. Think of line y=x, then y=-x and see how they're different.

TRUE OR FALSE: If no elements of a subspace pass through the origin of a graph, it is still a subspace

FALSE: If no elements of a subspace pass through the origin of a graph, then the subspace does not contain the zero vector which is one of the axioms of a vector space, and consequently a subspace too.

TRUE OR FALSE: A vector space must contain at least 2 vectors

FALSE: Nowhere in the definition does it specify the amount of vectors, other than the fact that it has to be non-empty. The vector space {0} is an example.

TRUE OR FALSE: If 𝐯 is a vector in a vector space V, and r and s are scalars such that r𝐯=s𝐯, then r=s

FALSE: Since the question asked for any vector, we can let 𝐯 be the zero vector. In this case r𝐯=s𝐯 will always be true since it will always =0, however, r doesn't necessarily equal s, they can be any scalar value and we will still get the same result.

TRUE OR FALSE: Three vectors x,y, and z in R3 always determine a 3-dimensional solid region in R3

FALSE: They need to be linearly independent to form a solid in r3.

TRUE OR FALSE: The set of positive real numbers, with the usual operations of addition and scalar multiplication, forms a vector space.

FALSE: This isn't even a scalar field since it specified it had to be positive, let alone a vector field since the elements of the set of positive real numbers are not vectors.

Linear Transformation

For two vectors, the following has to be true: 1. T(u+v) = T(u)+T(v) 2. T(cv) = cT(v) where u and v are vectors and c is a scalar

How does multiplying a matrix (with a determinant of 0) to a vector, transform that vector?

If a vector is being multiplied by a matrix with a determinant of 0, it causes one the unit vectors of the vector to point in the same direction as another one of the unit vectors, so the vector loses a dimension. The determinant is the factor by which a rectangular shape of n dimensions with all dimensions being 1 (a basic unit vector for all dimensions) is scaled, so for 2 dimensions, a two dimensional rectangular shape is just a rectangle with it's 𝐢 unit vector (vector of x coordinate) having a magnitude of 1, and it's 𝐣 unit vector (vector of y coordinate) having a magnitude of 1 as well. This creates a rectangle (yes it's also a square) with dimensions 1x1, so it's area is 1. If we multiplied this vector with components <i,j> by a matrix with a determinant of 5, it would transform the rectangle created by the two components to a new shape, a parallelogram, so there's two different ways it would have effected it, first way is that the x lengths of the parallelogram are 5, and the y lengths are 1 which would give the area 5, or the y lengths are 5 and the x lengths are 1. For 3 dimensions it would be the volume of a parallelepiped, and the determinant would just change the shape's volume not area. So now with that concept in mind, if we have a vector in R^2 (a 2 dimensional vector), multiplying a vector with components <i,j>, by a matrix with a determinant of 0, is saying that we are multiplying the shape with dimensions 1x1 by a factor of 0, which means we have a new area of 0, now think about what shape gives us an area of 0. It's not really a shape it's a line, so to make a line with two vectors, they have to lie on the same line, and so the shape loses a dimension because it drops from 2d to 1d (rectangle to a line).

Spanned/generated

If every vector in a vector space V, can be written as a linear combination of v1,v2,...,v_k, then V is spanned or generated by v1,v2,...,v_k and the set of vectors that we used in our linear combination, {v1,v2,..,v_k} is called the spanning set for V, or that {v1,v2,....,v_k} spans V.

How can you tell if a matrix is invertible?

If the determinant is non-zero

Symmetric Matrix

If the matrix is equal to it's transpose, it is said to be skew symmetric. All elements above the main diagonal of a symmetric matrix are reflected into equal elements, below the main diagonal. A= A^T

Skew Symmetric Matrix

If the matrix is equal to the negative version of it's transpose, it is said to be skew symmetric. All elements above the main diagonal of a skew symmetric matrix are reflected into the negative version of itself, below the diagonal. A= -A^T

Vector Space

If we have a set of vectors V and a scalar field F, V is a vector space over F, if the list of axioms are met, but the two most important are: 1. Closure under addition: For any pair of vectors 𝐮,𝐯 in V, then 𝐮+𝐯 must also exist somewhere in V, if the sum is not in V, then V is not closed under addition, and it is not a vector space over F. 2. Closure under scalar multiplication: For any vector 𝐮 in V, and any scalar k in F (a scalar field), then k𝐮 must also exist somewhere in V. If it is not contained somewhere in V it is not closed under scalar multiplication, and it is not a vector space over F.

Subspace

Let S be a nonempty subset of a vector space V. If S is itself a vector space under the same operations of addition and scalar multiplication as used in V, then we say that S is a subspace of V. S must be closed under addition and scalar multiplication. So you must be able to derive the original form in some way, even after addition or scalar multiplication.

If the rank of a matrix that represents a set of vectors (each column is a vector) is less than the number of columns, is the set linearly dependent or linearly independent?

Linearly dependent

If there is a free variable in the vector 𝐗 for the equation A𝐗 = 𝐁, is the set of vectors linearly independent or linearly dependent?

Linearly dependent, since k can be any number, meaning that there is an infinite number of solutions, and for a set to be linearly independent the only solution is the trivial solution.

Scalar Multiplication

Multiplying a matrix by a constant.

Will 3 coplanar vectors span R^3?

No, if they lie on the same plane there's an infinite set of vectors in R^3 that they are not touching because they are limited to a 2D plane even though we are in 3D space. So there's no linear combination that exists that will let us span R^3 if our 3 vectors are all coplanar.

Rank

Number of dimensions in the column space

Set notation

S ={𝘃 (is an element of) V: conditions of 𝘃}

Vectors in R^n

Set of all ordered n-tuples of real numbers

Null Space

Set of vectors that land on the origin (zero vector). When a vector is transformed, and it's rank becomes less than the number of columns, there's a set of vectors that get smooshed onto the origin. For A𝐱=𝐛, when 𝐛= the zero vector, A𝐱=zero vector, the null space gives you all the possible solutions to the equation. (Solution set to a homogeneous linear system A𝐱=0 is the null space of A or nullspace(A))

Span - another definition

Span of a set of vectors, is the set of all of their linear combinations. Basically, what are all the possible new vectors you can make, by multiplying the vectors in a set by some constants, and then adding all the vectors to find a point in space, which the new vector will point to.

Column Space

Span of the columns of a matrix. Also tells us if a solution exists or not.

TRUE OR FALSE: Any two vectors in R^2 that are not scalar multiples of each other will span all of R^2

TRUE

TRUE OR FALSE: For every vector (x1, x2, . . . , xn) in Rn, the vector (−1) · (x1, x2, . . . , xn) is an additive inverse.

TRUE

TRUE OR FALSE: The additive inverse of a vector v in a vector space V is unique

TRUE

TRUE OR FALSE: The vector 5i - 6j + root(2)k in R3 is the same as (5,-6,root(2))

TRUE

TRUE OR FALSE: The zero vector in a vector space V is unique.

TRUE

TRUE OR FALSE: For every vector x in Rn, the vector 0x is the zero vector of Rn

TRUE: Any vector multiplied by a factor of 0, makes the vector a zero vector.

TRUE OR FALSE: Any set of 3 vectors in R^2, is linearly dependent

TRUE: Since any pair of vectors in R^2, can linearly combine and make any other vector in R^2, then the 3rd vector is redundant (since it just is that linear combination of the other two vectors) so the set is linearly dependent.

TRUE OR FALSE: If 𝐯 is a 𝐧𝐨𝐧𝐳𝐞𝐫𝐨 vector in a vector space V, and r and s are scalars such that r𝐯=s𝐯, then r=s

TRUE: Since the question 𝐝𝐨𝐞𝐬 specify that the vector 𝐯 has to be nonzero, then there only exists one solution to r𝐯=s𝐯 and that is when r = s

TRUE OR FALSE: Each vector (x, y, z) in R3 has exactly one additive inverse.

TRUE: Think about a column vector, [x,y,z], I know that's not a column vector but just imagine it is with x as the first element, y as the second, and z is the bottom, then there exists [-x,-y,-z], that you can add to the original column vector that will give you [0,0,0]

TRUE OR FALSE: Every pair of numbers gives you one and only one vector. And every vector is associated with one and only one pair of numbers. (this is for 2 dimensions. In 3 dimensions it would be every triplet of numbers)

TRUE: Think about it, there is only one way to get to a point with a straight line (the vector) from the origin.

TRUE OR FALSE: If 𝐱 and 𝐲 are vectors in a vector space V, then the additive inverse of 𝐱 + 𝐲 is (-𝐱) + (-𝐲).

TRUE: Think about it, we want (𝐱 + 𝐲) + (something else) = 0, we know the additive inverse of a vector is -𝐱 if our vector is 𝐱 and -𝐲 if our vector is 𝐲. So the (something else) = -(𝐱 + 𝐲). By the commutative, associative, additive inverse, and distributive laws we know this will give us 0 making it the additive inverse.

TRUE OR FALSE: The set {0}, with the usual operations of addition and scalar multiplication, forms a vector space

TRUE: This is the simplest vector space, it is a set of a vector {0} (the zero vector), that is closed under addition and multiplication over any scalar field, which is why the scalar field doesn't even need to be defined (in any other instance it would need to be).

TRUE OR FALSE: The solution set to a linear system of 4 equations and 6 unknowns consists of a collection of vectors in R6

TRUE: however there will be free variables

Components

The coordinates of a vector.

Linear Combination

The sum of a set of vectors {v1,v2,v3...v_n} in a vector space V, multiplied by a set of scalar values {a1,a2,a3....an}, which will then give you a resulting vector {a1*v1 + a2*v2 +a3*v3 +.....+ an*v_n}. For example: if we have a set of vectors {v1, v2,v3}, and a set of scalars {a1,a2,a3} where a1=4, a2 =6, a3=12. The linear combination is a vector B that equals = (4*v1 + 6*v2 + 12*v3), which will simplify down to one vector that is also contained in V.

Complex Vector Space

The vector space over the complex numbers. Any set of vectors V over the scalar field of complex numbers C.

Real Vector Space

The vector space over the real numbers. Any set of vectors V over the scalar field of real numbers R.

Zero vector in column space

The zero vector is always included in the column space

TRUE OR FALSE: If x and y are vectors in R2 whose components are even integers and k is a scalar, then x+y and kx are also vectors in R2 whose components are even integers

True: For the first case x+y, two even numbers added together still gives you an even number, and for the second case kx, a number multiplied by an number is still an even number

Non-homogeneous system of linear equations

When a system of linear equations has a unique non-trivial solution.

Homogeneous system of linear equations

When all of the constant terms of a system of linear equations are 0. One way to tell if a system is homogeneous, is to look at the equation AX = B representing a system of linear equations, where the vector B, is also the zero vector.

Full Rank

When the rank = the number of columns it is full rank.

Trivial Solution

When the resulting vector, to the solution of a system of equations, has all of its coordinates equaling 0 (When the solution is the zero vector).