Linear Algebra: Matrix Basics
Define: Matrix
- A rectangular array of numbers that can correspond to various things. - Alternatively, it is a collection of row and column vectors
Define: ECHELON form (not reduced!)?
1) Nonzero rows above 0 rows 2) The leading entry of a row is above and to the left of the leading entry below it. 3) Only 0s are below a leading entry column
For the Matrix Solution: What is the parametric form?
1) Write solutions to each variable in column, in terms of free variables (if any) and constants. 2) Divide each free variable into its own respective column vector from top to bottom, and all constants in their own column vector. 3) For the free variable columns, factor out the variable as a scalar. 4) Write as a Linear Combination
For a Matrix: (A+B) + C = ?
A + (B + C) associative property of addition holds. Having the same dimensions are the key!
Define: Pivot Positions and Pivot Columns
A Pivot position is a spot that corresponds to where a leading entry goes. A pivot column is just a column containing one.
Matrix Equation Form
A is the coefficient Matrix of the system x is the column vector that contains each variable in the system b is the column vector that each equation equals Multiply A*x across then down to align each variable with respective coefficient and get the system back.
Define: Square matrix?
A matrix in which m = n n x n That is the rows equal the columns.
Define: Identity matrix?
A square matrix composed of 0s and 1s. With the 1s along the diagonal of the matrix. Note the subscript on the I which denotes the size of the matrix.
IA = ? AI = ?
A. The identity matrix gives the same matrix it is being multiplied by and is commutative.
Skew Matrix
A^T = -A Special Matrix
Symmetric Matrix
A^T = A Special Matrix
Augmented Matrix?
Ax = b A coefficient matrix with the addition of the b constants column in the RIGHTMOST position.
For a Matrix: A + B = ?
B + A commutative property of addition holds. Having the same dimensions are the key!
How is the inverse found using row reduction?
Essentially you are transforming the left-sided matrix A into the identity matrix using row reduction.
A⁻¹A =? AA⁻¹ = ?
I. An inverse of a matrix multiplied by the matrix gives the identity matrix and is communtative.
Define: REDUCED row echelon form (RRE)?
In addition to echelon form requirements: - Each leading entry is 1 - Each leading 1 is the only nonzero in the column.
Is matrix multiplication commutative?
In general, no, with a a few exceptions. The dimensions are the key reason. AB ≠ BA
How is matrix multiplication done?
It's the dot product of the: - row of the first matrix - with the column of the second matrix across then down mnemonic
How are Matrices added/subtracted?
Just add/subtract each corresponding entry. They must therefore have the same dimensions in order to be added/subtracted
What do the rows and columns of a matrix represent?
Rows: Number of equations in the system Columns: Number of variables/unknowns (not counting the last column in a augmented matrix)
Coefficient Matrix
Simply the coefficients of the system of equations. NOT INCLUDING WHAT THEY ARE EQUAL TO
Define: Singular and Non-singular matrices?
Singular: a matrix that has no inverse Nonsingular: a matrix with an inverse
How can you use determinants to find the inverse of a 2x2 matrix?
So if the determinant is 0, that means it's a singular matrix.
Upper-Triangular Matrix
Special Matrix where all non-zero values are ABOVE the diagonal line.
Lower-Triangular Matrix
Special Matrix where all non-zero values are BELOW the diagonal line.
Diagonal Matrix
Special Matrix where the only non-zero entries in the matrix are along the diagonal line.
What are the requirements for matrices to be multiplied?
The columns of the first matrix must equal the rows of the second matrix. In other words, the inner dimensions of the matrices must match.
Define: Leading Entry?
The leftmost nonzero entry in a row
For Matrix Multiplication: What will be the dimensions of the product matrix?
The product will have the rows of the first matrix the columns of the second matrix. In other words, the outer dimensions of the original matrices are the product's dimensions.
What is the transpose of a matrix and how is it denoted?
You just swap rows with columns.
How are matrix entries denoted?
a is the matrix i is the row j is the column EX: a₁₂ is the entry in row 1 column 2
Scalar multiplication of a matrix?
multiplies each entry of the matrix
row and column vectors?
row = 1 x n column = m x 1 Matrices consisting of 1 row or column respectively
Matrix Dimensions are given by?
rows x columns m x n