Linear algebra

• 
  

  Linear algebra

• Vectors and spaces
• Matrix transformations
• Alternate coordinate systems (bases)

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Vectors and spaces

Let's get our feet wet by thinking in terms of vectors and spaces.

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Vectors

We will begin our journey through linear algebra by defining and conceptualizing what a vector is (rather than starting with matrices and matrix operations like in a more basic algebra course) and defining some basic operations (like addition, subtraction and scalar multiplication).

• Introduction to Vectors
• Vector Examples
• Scaling vectors
• Adding vectors
• Parametric Representations of Lines

Linear combinations and spans

Given a set of vectors, what other vectors can you create by adding and/or subtracting scalar multiples of those vectors. The set of vectors that you can create through these linear combinations of the original set is called the "span" of the set.

• Linear Combinations and Span

Linear dependence and independence

If no vector in a set can be created from a linear combination of the other vectors in the set, then we say that the set in linearly independent. Linearly independent sets are great because there aren't any extra, unnecessary vectors lying around in the set. :)

• Introduction to Linear Independence
• More on linear independence
• Span and Linear Independence Example

Subspaces and the basis for a subspace

In this tutorial, we'll define what a "subspace" is --essentially a subset of vectors that has some special properties. We'll then think of a set of vectors that can most efficiently be use to construct a subspace which we will call a "basis".

• Linear Subspaces
• Basis of a Subspace

Vector dot and cross products

In this tutorial, we define two ways to "multiply" vectors-- the dot product and the cross product. As we progress, we'll get an intuitive feel for their meaning, how they can used and how the two vector products relate to each other.

• Vector Dot Product and Vector Length
• Proving Vector Dot Product Properties
• Proof of the Cauchy-Schwarz Inequality
• Vector Triangle Inequality
• Defining the angle between vectors
• Defining a plane in R3 with a point and normal vector
• Cross Product Introduction
• Proof: Relationship between cross product and sin of angle
• Dot and Cross Product Comparison/Intuition
• Vector Triple Product Expansion (very optional)
• Normal vector from plane equation
• Point distance to plane
• Distance Between Planes

Matrices for solving systems by elimination

This tutorial is a bit of an excursion back to you Algebra II days when you first solved systems of equations (and possibly used matrices to do so). In this tutorial, we did a bit deeper than you may have then, with emphasis on valid row operations and getting a matrix into reduced row echelon form.

• Matrices: Reduced Row Echelon Form 1
• Matrices: Reduced Row Echelon Form 2
• Matrices: Reduced Row Echelon Form 3

Null space and column space

We will define matrix-vector multiplication and think about the set of vectors that satisfy Ax=0 for a given matrix A (this is the null space of A). We then proceed to think about the linear combinations of the columns of a matrix (column space). Both of these ideas help us think the possible solutions to the Matrix-vector equation Ax=b.

• Matrix Vector Products
• Introduction to the Null Space of a Matrix
• Null Space 2: Calculating the null space of a matrix
• Null Space 3: Relation to Linear Independence
• Column Space of a Matrix
• Null Space and Column Space Basis
• Visualizing a Column Space as a Plane in R3
• Proof: Any subspace basis has same number of elements
• Dimension of the Null Space or Nullity
• Dimension of the Column Space or Rank
• Showing relation between basis cols and pivot cols
• Showing that the candidate basis does span C(A)

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Matrix transformations

Understanding how we can map one set of vectors to another set. Matrices used to define linear transformations.

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Functions and linear transformations

People have been telling you forever that linear algebra and matrices are useful for modeling, simulations and computer graphics, but it has been a little non-obvious. This tutorial will start to draw the lines by re-introducing you functions (a bit more rigor than you may remember from high school) and linear functions/transformations in particular.

• A more formal understanding of functions
• Vector Transformations
• Linear Transformations
• Matrix Vector Products as Linear Transformations
• Linear Transformations as Matrix Vector Products
• Image of a subset under a transformation
• im(T): Image of a Transformation
• Preimage of a set
• Preimage and Kernel Example
• Sums and Scalar Multiples of Linear Transformations
• More on Matrix Addition and Scalar Multiplication

Linear transformation examples

In this tutorial, we do several examples of actually constructing transformation matrices. Very useful if you've got some actual transforming to do (especially scaling, rotating and projecting) ;)

• Linear Transformation Examples: Scaling and Reflections
• Linear Transformation Examples: Rotations in R2
• Rotation in R3 around the X-axis
• Unit Vectors
• Introduction to Projections
• Expressing a Projection on to a line as a Matrix Vector prod

Transformations and matrix multiplication

You probably remember how to multiply matrices from high school, but didn't know why or what it represented. This tutorial will address this. You'll see that multiplying two matrices can be view as the composition of linear transformations.

• Compositions of Linear Transformations 1
• Compositions of Linear Transformations 2
• Matrix Product Examples
• Matrix Product Associativity
• Distributive Property of Matrix Products

Inverse functions and transformations

You can use a transformation/function to map from one set to another, but can you invert it? In other words, is there a function/transformation that given the output of the original mapping, can output the original input (this is much clearer with diagrams). This tutorial addresses this question in a linear algebra context. Since matrices can represent linear transformations, we're going to spend a lot of time thinking about matrices that represent the inverse transformation.

• Introduction to the inverse of a function
• Proof: Invertibility implies a unique solution to f(x)=y
• Surjective (onto) and Injective (one-to-one) functions
• Relating invertibility to being onto and one-to-one
• Determining whether a transformation is onto
• Exploring the solution set of Ax=b
• Matrix condition for one-to-one trans
• Simplifying conditions for invertibility
• Showing that Inverses are Linear

Finding inverses and determinants

We've talked a lot about inverse transformations abstractly in the last tutorial. Now, we're ready to actually compute inverses. We start from "documenting" the row operations to get a matrix into reduced row echelon form and use this to come up with the formula for the inverse of a 2x2 matrix. After this we define a determinant for 2x2, 3x3 and nxn matrices.

• Deriving a method for determining inverses
• Example of Finding Matrix Inverse
• Formula for 2x2 inverse
• 3x3 Determinant
• nxn Determinant
• Determinants along other rows/cols
• Rule of Sarrus of Determinants

More determinant depth

In the last tutorial on matrix inverses, we first defined what a determinant is and gave several examples of computing them. In this tutorial we go deeper. We will explore what happens to the determinant under several circumstances and conceptualize it in several ways.

• Determinant when row multiplied by scalar
• (correction) scalar multiplication of row
• Determinant when row is added
• Duplicate Row Determinant
• Determinant after row operations
• Upper Triangular Determinant
• Simpler 4x4 determinant
• Determinant and area of a parallelogram
• Determinant as Scaling Factor

Transpose of a matrix

We now explore what happens when you switch the rows and columns of a matrix!

• Transpose of a Matrix
• Determinant of Transpose
• Transpose of a Matrix Product
• Transposes of sums and inverses
• Transpose of a Vector
• Rowspace and Left Nullspace
• Visualizations of Left Nullspace and Rowspace
• Rank(A) = Rank(transpose of A)
• Showing that A-transpose x A is invertible

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Alternate coordinate systems (bases)

We explore creating and moving between various coordinate systems.

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Orthogonal complements

We will know explore the set of vectors that is orthogonal to every vector in a second set (this is the second set's orthogonal complement).

• Orthogonal Complements
• dim(V) + dim(orthogonal complement of V)=n
• Representing vectors in Rn using subspace members
• Orthogonal Complement of the Orthogonal Complement
• Orthogonal Complement of the Nullspace
• Unique rowspace solution to Ax=b
• Rowspace Solution to Ax=b example

Orthogonal projections

This is one of those tutorials that bring many ideas we've been building together into something applicable. Orthogonal projections (which can sometimes be conceptualized as a "vector's shadow" on a subspace if the light source is above it) can be used in fields varying from computer graphics and statistics! If you're familiar with orthogonal complements, then you're ready for this tutorial!

• Projections onto Subspaces
• Visualizing a projection onto a plane
• A Projection onto a Subspace is a Linear Transforma
• Subspace Projection Matrix Example
• Another Example of a Projection Matrix
• Projection is closest vector in subspace
• Least Squares Approximation
• Least Squares Examples
• Another Least Squares Example

Change of basis

Finding a coordinate system boring. Even worse, does it make certain transformations difficult (especially transformations that you have to do over and over and over again)? Well, we have the tool for you: change your coordinate system to one that you like more. Sound strange? Watch this tutorial and it will be less so. Have fun!

• Coordinates with Respect to a Basis
• Change of Basis Matrix
• Invertible Change of Basis Matrix
• Transformation Matrix with Respect to a Basis
• Alternate Basis Transformation Matrix Example
• Alternate Basis Transformation Matrix Example Part 2
• Changing coordinate systems to help find a transformation matrix

Orthonormal bases and the Gram-Schmidt Process

As we'll see in this tutorial, it is hard not to love a basis where all the vectors are orthogonal to each other and each have length 1 (hey, this sounds pretty much like some coordinate systems you've known for a long time!). We explore these orthonormal bases in some depth and also give you a great tool for creating them: the Gram-Schmidt Process (which would also be a great name for a band).

• Introduction to Orthonormal Bases
• Coordinates with respect to orthonormal bases
• Projections onto subspaces with orthonormal bases
• Finding projection onto subspace with orthonormal basis example
• Example using orthogonal change-of-basis matrix to find transformation matrix
• Orthogonal matrices preserve angles and lengths
• The Gram-Schmidt Process
• Gram-Schmidt Process Example
• Gram-Schmidt example with 3 basis vectors

Eigen-everything

Eigenvectors, eigenvalues, eigenspaces! We will not stop with the "eigens"! Seriously though, eigen-everythings have many applications including finding "good" bases for a transformation (yes, "good" is a technical term in this context).

• Introduction to Eigenvalues and Eigenvectors
• Proof of formula for determining Eigenvalues
• Example solving for the eigenvalues of a 2x2 matrix
• Finding Eigenvectors and Eigenspaces example
• Eigenvalues of a 3x3 matrix
• Eigenvectors and Eigenspaces for a 3x3 matrix
• Showing that an eigenbasis makes for good coordinate systems