M2 Burzcak Week 1
What are minors for matrices?
A "minor" is the determinant of the square matrix formed by deleting one row and one column from some larger square matrix. Since there are lots of rows and columns in the original matrix, you can make lots of minors from it. These minors are labeled according to the row and column you deleted. Minors are used to calculate determinants for matrices of 4x4 size and bigger. Technically, you're "supposed" to go down to 2-by-2 determinants when you "expand" by this method but 3x3 is faster. Also, If you add a multiple of one row (or column) to another row (or column), the value of the determinant will not change. In other words, you can do row operations on determinants, creating a row (or column) with lots of zeroes, and you'll still get the right answer. (You can also just multiply rows -- without the adding -- or switch rows, but those operations will change the determinant's value. The changes are annoying to keep track of, so try only to do the row-addition operation.)
What is an affine space? (Math Stack exchange)
Consider an infinite sheet (of idealized paper, if you like). If it is blank, then there is absolutely no way to distinguish between any two points on the sheet. Nonetheless, if you do have two points on the sheet, you can measure the distance between them. And if there is a uniform magnetic field parallel to the sheet, then you can even measure the bearing from one point to another. Thus, given any point PP on the sheet, you can uniquely describe every other point on the sheet by its distance and bearing from PP; and conversely, given any distance and bearing, there is a point with that distance and bearing from PP. This is the situation that the notion of a 2-dimensional affine space is an abstraction of. Now suppose we have marked a point OO on the sheet. Then we can "add" points PP and QQ on the sheet by drawing the usual parallelogram diagram. The result P+QP+Q of the "addition" depends on the choice of OO (and, of course, PP and QQ), but nothing else. This is what the notion of a 2-dimensional vector space is an abstraction of. Another explanation: Consider the vector space R3R3. Inside R3R3 we can choose two planes, P1P1 and P2P2. The plane P1P1 passes through the origin but the plane P2P2 does not. It is a standard homework exercise in linear algebra to show that the P1P1 is a sub-vector space of R3R3 but the plane P2P2 is not. However, the plane P2P2 resembles a 22-dimensional vector space in many ways, primarily in that it exhibits a linear structure. In fact, P2P2 is a classical example of an affine space. One defect of the plane P2P2 is that it has no distinguished origin. One can artificially choose a point and redefine the algebraic operations in such a way to give it an origin, but that is not inherent to P2P2. Another problem is that the sum of two vectors in P2P2 is no longer in P2P2. One can think of AR2AR2 as being modeled on this situation
Cramer's rule?
Given a system of linear equations, Cramer's Rule is a handy way to solve for just one of the variables without having to solve the whole system of equations. We have the left-hand side of the system with the variables (the "coefficient matrix") and the right-hand side with the answer values. Let D be the determinant of the coefficient matrix of the above system, and let Dx be the determinant formed by replacing the x-column values with the answer-column values: If D=o, can't use Cramer's rule.
What is the Cauchy-Binet Formula?
If m = n - det(AB)=det(A)det(B) That is, the determinant of the product is equal to the product of the determinants. if m > n -det(AB)=0
What is linear span?
In linear algebra, the linear span of a set S of vectors in a vector space is the smallest linear subspace that contains the set. It can be characterized either as the intersection of all linear subspaces that contain S, or as the set of linear combinations of elements of S. The linear span of a set of vectors is therefore a vector space
What is the rank of a matrix?
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by . There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by or ; sometimes the parentheses are not written, as in. Or more simply: The number of non-zero rows is the matrice's rank.
What is linear (vector) basis?
In mathematics, a set B of elements (vectors) in a vector space V is called a basis, if every element of V may be written in a unique way as a (finite) linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates on B of the vector. The elements of a basis are called basis vectors. Equivalently B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B. In more general terms, a basis is a linearly independent spanning set. A vector space can have several bases; however, all the bases have the same number of elements, called the dimension of the vector space. More simply put, vector basis is for a set of vector V are the linearly independent vectors (i.e there aren't too many of them) and span the vector space enough to read all the vectors.
What is an orthogonal basis?
In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal. - An orthogonal set of non-zero vectors is linearly independent.
What is Dimension (in Vector space)?
In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined.
How to get the inverse of a Matrix using Minors, Cofactors and Adjugate? (Laplace Expansion)
In step 4, we use the determinant of the original matrix.
Definition of Linear independence?
In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent.
What is the triple scalar product?
It's the determinant of the matrix spawned by the components of the vectors (components come in rows - not columns and follow ordre of vectors from left to right)
What is an orthonormal basis?
Like orthogonal but the basis has a magnitude of 1.
What does the rank of matrices tell us?
M is the matrix without the solution (one extra column) A is the matrix which includes the solutions. (b or however they are denoted. (Then M has m rows (corresponding to m equations) and n columns (corresponding to n unknowns) and A has one more column (the constants). The following summary outlines the possible cases.)
What is the adjugate for matrices?
One needs to find the cofactors of a matrix, write them down as a matrix and finally transpose.
What does an only 0 row tells us?
That either the equations are inconsistent or there are infinitely many solutions. 0 rows also indicate that this row (vector) is linearly dependent on all the others.
How to find the inverse of a 2x2 Matrix?
The inverse is like division but for matrices. First of all, to have an inverse the matrix must be square. Also, the determinant cannot be zero (or we end up dividing by zero). Also order is important: - AX = B --> X = (A^-1)B - XA = B --> X = B(A-1)
How to row reduce a matrix (elementary row operations).?
Use the first row to clear the rest of the first column; use the new secondrow to clear the rest of the second column; etc. Also, since matrices are equal onlyif they are identical, we will not use equal signs between them.
How to find the determinent of a matrix of size 4x4 and bigger?
Weird fact: It doesn't matter which row or column you use for your expansion; you'll get the same value regardless. But this flexibility can be useful.
Inverse of a Matrix using Elementary Row Operations?
Write an identity next to the original matrix. Try to make the original matrix into an identity one but apply all the steps on the identity matrix. When it is done, our original identity matrix will be the inverse matrix.
Inverse of a 3x3 matrix? (Fastest way)
https://youtu.be/C7D36h_0Zlw Video is good as it shows a fast way to do Adjoint as well.
Matrix Multiplication?
- To multiply an m×n matrix by an n×p matrix, the n's must be the same,and the result is an m×p matrix. - matrix multiplication is not commutative (Except for the matrix identity) Order matters!