- [[Vectors]]:
- Mathematical objects representing quantities with both magnitude and direction, often expressed in coordinate form.
- [[Linear combinations and spans]]
- A linear combination involves summing scaled versions of vectors;
- span is the set of all possible linear combinations of a given set of vectors.
- [[Programmimg/Math/Subspaces and the basis for a subspace|Subspaces and the basis for a subspace]]
- A subspace is a set of vectors closed under vector addition and scalar multiplication.
- A basis is a minimal set of vectors that span the subspace.
- [[Linear Dependance and Independence]]
- Linear Independence: No vector can be expressed as a combination of others. (unique solution exists). There is not noise.
- Linear Dependence: At least one vector can be represented as a combination of others. Infinitely many solutions or no solution may exist.
- Crammer Method (2x2) (3x3):
- A method using determinants to solve systems of linear equations. It works well for small systems but is inefficient for larger ones. If the determinant == 0, the system has no unique solution.
- [[Gaussian Elimination]]:
- A systematic method for simplifying systems of linear equations through row operations to reach an upper triangular form, facilitating back substitution.
- [[Gauss-Jordan Elimination]]:
- An extension of Gaussian elimination that transforms the matrix to reduced row echelon form (RREF) ( results to [[I - Identity Matrix]]), allowing for direct reading of solutions without back substitution.
- [[Matrix Vector Multiplication]]
- linear transformation by multiplication.
- [[Basis for the Column Space of a Matrix]]
- A set of linearly independent column vectors in a matrix that spans the column space, crucial for understanding the matrix's range.
- [[Inverse Matrix]]inverse is just like identity divided by matrix
$A \cdot A^{-1} = A^{-1} \cdot A = I$ -
Adjugate and Determinant Method:
- (2x2): $A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$
-
(>3x3):
$\text{det}(E) = a(ei - fh) - b(di - fg) + c(dh - eg)$
- [[Gauss-Jordan Elimination]]:
- Augmented Matrix Transformation:
$[U ∣ I]$ into the form$[I ∣ U^{-1}]$ - Why it works? : let say $U=2$, $I=1$, what is $U^{-1}$?
-
$2 | 1$ =>$\frac{R1}{2}$ =>$\frac{2}{2}$ =>$1|0.5$ $2*0.5=1$
-
- Augmented Matrix Transformation: