Pure Mathematics C1: Linear Equations & Matrices

80 flashcards · Mathematics · by lakesare

Name all 3 components here.
How would the system of m linear equations in n unknowns look?
What's the format of the solution set of the system of linear equations?
{ (0, 1, 0), (5, 5, 3), ~ }
When is the system of linear equations consistent?
Has at least 1 solution. Or, equivalently, there exists the interpretation under which all sentences are true.
What's the format of the solution set when the system of linear equations is infinite?
{(k, 1 − k, 0) : k ∈ R}
When is the system of linear equations homogeneous?
When all constant terms are 0.
When it comes to homogeneous systems of linear equations - what's the trivial solution?
(0, 0, 0)
Name all 3 solutions any system of linear equations can possibly have.
1 solution 0 solutions ∞ solutions
Explain 3 elementary operations. ra ↔ rb ra → αra ra → ra + βrb 3 elementary operations represented in symbols, where α, β are non-zero numbers.
ra ↔ rb - interchange 2 equations ra → αra - multiply equation ra → ra + βrb - add multiple of another equation We label the equations r1, r2, etc. (r stand for row).
What's the orderly algorithm of the Gauss-Jordan elimination?
Use 1st equation to eliminate 1st unknown from all eqs. Use 2nd equation to eliminate 2nd unknown from all eqs. Etc.
What's the goal form of the Gauss-Jordan elimination?
[Visualizing] How to visualize the system of linear equations?
Dimensionality = the number of distinct unknowns (3 in our case) Each equation: in 3d - plane in 2d - line in 1d - point The solution set is equation ∩ equation ∩ equation.
[Visualizing] What's the issue with visualizing this system of equations in 3d? The same system of linear equations in 2d.
We get the ∞ solutions because z is free to wander! In reality we should only get 1 solution.
What's m and n?
m rows, n columns.
Imagine the 0 matrix.
What's the additive identity in Mm,n?
0m,n
(Mm,n, +) is a group. What's Mm,n here?
All matrices of the dimension m×n.
Visually, what's the result?
Visually imagine the diagonal matrix.
Visually imagine the In.
Visually imagine the result.
How do you read AT?
A transpose.
What's the official definition of taking the transpose AT?
(i, j) entry becomes the (j, i) entry.
What is this matrix transpose property similar to?
Name 2 steps to turn this system of linear equations into Ax = b.
What do we call this?
Matrix multiplication is distributive over matrix addition. Because multiplication kindly distributes itself to each of the added numbers.
[Uniqueness of the inverse] How would you use associativity to prove that if A has the inverse B, and if A has the inverse C, then B = C.
Wrap A with its inverses from both sides: BAC = (BA)C = B(AC).
Visually imagine the AT of the square matrix.
Visually imagine the AT of this matrix.
How are symmetric matrices connected to transposition?
The matrix is symmetric iff AT = A. Thus only square matrices can be symmetric.
What do we usually call: the additive inverse of A (–A) the multiplicative inverse of A (B such that AB=I=BA)
additive inverse - negative of A multiplicative inverse - inverse of A
Exactly what "inverse" are they talking about?
Multiplicative inverse.
What's meant by invertible matrix?
The matrix that has a multiplicative inverse.
[Groups and matrices] Name 2 groups that matrices can form.
(Mm,n, +) forms a group (Mn,n invertible, *) forms a group
What's a good way to prove that something is an inverse of something?
Apply the AB = I = BA definition! –1 is not some operation you can perform to find the inverse.
How would you solve it with a system of linear equations?
Use the equation of the plane ax + by + cz = d.
Leading entry VS leading 1?
leading entry - first non-zero entry E.g. –5 or 1 in the picture.leading 1 - first non-zero entry that is 1 E.g. 1 in the picture.
What's the augmented matrix of this system of linear equations?
What's the system of linear equations corresponding to this augmented matrix? Write d own t he system of linear equations corresponding to the following augmented matrix, given that the unknowns are, in order, x1, x2.
Why did we introduce the augmented matrix of the system of linear equations altogether?
Just easier to use the abbreviated form (the augmented matrix) rather than the full equations!
[Checking your solution] What's the gist of the row-sum check?
Create the row-sum as if it's an extra column At every step, make sure the sum still works!
What's this form of the matrix called?
Row-reduced matrix.
[Solutions in row-reduced form] How many solutions does this row-reduced matrix have?
1 solution.
[Solutions in row-reduced form] How many solutions does this row-reduced matrix have?
0 solutions.
[Solutions in row-reduced form] How many solutions does this row-reduced matrix have?
∞ solutions. We want to express each leading unknown in terms of non-leading unknowns (superseding each of those with x3 = k, k∈R).
Name 3 steps to write down the solution. Consider you have this row-reduced augmented matrix. Assume that the unknowns are x1, x2, x3, x4, x5.
Write as a system of linear equations. Each leading unknown = ~non-leading unknowns~. Each non-leading unknown = x3 = k, k∈R
[Row-reduction] What do you do in this situation (first 0)?
Interchange r1 ↔ r2.
[Row-reduction] Imagine visually how would you row-reduce this matrix.
Augmented matrix VS matrix form of the system of linear equations?
augmented matrix - abbreviated notation for the system of linear equations. matrix form - Ax = b, actual mathematical expression.
[Finding the inverse of the matrix] What does this scheme denote?
To find the matrix inverse, we should 1. row reduce our matrix, and 2. apply same operations to I in parallel.
[Finding the inverse of the matrix] What's a shorter way to write this strategy for finding the inverse?
[Finding the inverse of the matrix] When is this strategy for finding the inverse most useful?
For larger matrices.
How do we know we can manipulate the equation Ax = b into x = A–1b?
Interesting way to show that these 3 statements are equivalent?
What's an elementary matrix?
Matrix obtained by performing 1 elementary row operation on I.
En is an elementary matrix. What's the result of Ek...E1A?
Ek...E1A = A with operation 1 applied to A, operation 2 applied to A, ...operation n applied to A. Multiplication on the left by an elementary matrix has the same effect as applying the associated elementary row operation.
What's an interesting connection between elementary row operations and its inverse, and elementary matrices A and B associated with them.
A is the actual matrix inverse of B too!
[Cramer's rule] What would the solution (using the Cramer's rule) look like? 2 equations in 2 unknowns.
[Cramer's rule] What's a Cramer's rule? Gabriel Cramer
Cramer's rule is an explicit formula for the solution of a system of linear equations with n equations and n unknowns.
What's another name for reduced row form of the matrix?
Reduced row echelon form. Echelon - a formation of troops, ships, aircraft, or vehicles in parallel rows with the end of each row projecting further than the one in front. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore, only row echelon forms are considered in the r…
What's a quick way to find the inverse of the 2x2 matrix?
[Geometric interpretation of the determinant] What determinant gives the area of this parallelogram?
(a, c) is one position vector, (b, d) is another position vector.
[Finding the determinant] What's the formula for the cofactor A23 of the 4x4 matrix? Cofactor Aij is the matrix associated with the entry aij of the matrix A.
[Finding the determinant] What's a submatrix?
Submatrix - matrix formed by removing some rows and/or columns.
[Finding the determinant] How do we denote the cofactor associated with the entry aij of the matrix A?
Aij.
[Finding the determinant] How to find the determinant of this huge matrix?
a11 - entry a11A11 - cofactor associated with the a11 entry
[Finding the determinant] What does it mean to "expand along the top row"?
To use the first row to find the determinant. a11 - entry a11A11 - cofactor associated with the a11 entry
What's an easier alternative to show that A is the inverse of B?
Simply show that AB = I.
[Finding the determinant] What's a11 and A11?
a11 - entry a11 A11 - cofactor associated with the entry a11
[Properties of matrices] det(AB) = ______________
det(AB) = det(A) det(B)
[Invertible matrix theorem] What does the Invertible Matrix Theorem basically claim?
There are 2 types of square matrices: invertible - for which all points are true! non-invertible - for which all points are false.
[Invertible matrix theorem] Name 2 points from the Invertible matrix theorem. Invertible matrix theorem lists all which are only true for invertible square matrices) A is invertible ______________ ______________ There are 2 types of square matrices:invertible - for which all points are true!non-invertible - for which all points are false.
det A ≠ 0 Ax = b has exactly 1 solution
[Matthew Bernstein blog post] What exactly is a "space"?
First a foremost - a set (with some structure).
[Matthew Bernstein blog post] When exactly is a space D-dimensional?
When we need exactly D pieces of information to describe each object in that space.
[Matthew Bernstein blog post] Intrinsic dimensionality VS extrinsic dimensionality?
intrinsic dimensionality - n of required pieces of information that we need to describe each object. extrinsic dimensionality - n of pieces of information we are using. Take the following example: we want to describe all points on a flat piece of paper in 3D space. That is, all of the points that we care about will lie only on the sheet of paper.Of course, we can use each points x, y, and z coordinates, but do we really need three pieces of information to describe each point on the paper? The a…
[Matthew Bernstein blog post] Visually imagine why a function behaves like a vector? Functions can also form vector spaces! The set of all continuous, real-valued functions forms a vector space.
f + g (vector + vector) Intuitively we see that such functions act like vectors in that we can add them together.cg (multiplication by scalar) We can also scale functions. In the following figure, the function g is scaled by c.
[Matthew Bernstein blog post] Visually imagine the 0 vector in the vector space consisting of functions. Functions can also form vector spaces! The set of all continuous, real-valued functions forms a vector space.
Lastly, the zero function acts as the zero vector. Here we depict the zero function, which outputs 0 for all inputs.
[Determinant VS inverse] Name the original way to find: determinant of the matrix |A| inverse of the matrix A–1
determinant of A - recursion (expand along the top row) inverse of A
[Determinant VS inverse] Name the short way to find: determinant of the matrix |A| inverse of the matrix A–1
determinant of A inverse of A
What's: transpose in linear algebra VS transposition in abstract algebra?
transpose in linear algebra transposition in abstract algebra