Purdue MA 265 Flashcards

Matrix Definitions and Properties

• A square matrix is singular if its reduced row echelon form (rref) contains a zero row or its determinant equals zero.
• An elementary matrix results from applying a single elementary row operation to the identity matrix.

Determinant Properties

• All elementary matrices are non-singular, meaning they have a non-zero determinant.
• The determinant of a matrix remains unchanged when transposed.
• Interchanging two rows of a matrix negates its determinant (det(A) becomes -det(A)).
• Multiplying a row by a scalar ( k ) scales the determinant by ( k ) (det(A) becomes k * det(A)).
• Adding a multiple of one row to another row leaves the determinant unchanged.
• If ( A ) and ( B ) are matrices, then det(AB) = det(A) * det(B).
• If det(A) is non-zero, matrix ( A ) is linearly independent.

Matrix Types

• A symmetric matrix satisfies the property ( A^T = A ).
• A skew-symmetric matrix satisfies ( A^T = -A ).

Cramer's Rule

• Used to solve systems of linear equations with a non-zero determinant.

Properties of Matrix Operations

• For matrix addition:
  • A + B = B + A (commutative),
  • A + (B + C) = (A + B) + C (associative),
  • There exists a zero matrix such that A + 0 = A,
  • Each matrix has an additive inverse where A + (-A) = 0.
• For matrix multiplication:
  • A(BC) = (AB)C (associative),
  • (A + B)C = AC + BC (distributive),
  • C(A + B) = CA + CB (distributive).

Properties of Transpose

• A double transpose returns the original matrix: ( (A^T)^T = A ).
• The transpose of a sum of matrices equals the sum of their transposes.
• The transpose of a product reverses the order: ( (AB)^T = B^T A^T ).
• The transpose of a scalar multiplied by a matrix is the scalar multiplied by the transpose of the matrix: ( (rA)^T = rA^T ).

Linear Independence and Basis

• Vectors ( V_1, \ldots, V_r ) are linearly dependent if there are constants ( a_1, a_2, \ldots, a_r ) (not all zero) such that ( a_1V_1 + a_2V_2 + \cdots + a_rV_r = 0 ).
• A set of ( n ) vectors in ( \mathbb{R}^n ) is linearly independent if the determinant of their corresponding matrix ( A ) is non-zero.
• In an ( n )-dimensional vector space ( V ):
  • Any subset of more than ( n ) vectors must be linearly dependent.
  • Any subset with fewer than ( n ) vectors cannot span ( V ).
• A finite set ( S ) of vectors in ( V ) with span equal to ( V ) has a maximal independent subset ( T ) that forms a basis for ( V ).
• If ( S ) spans ( V ) or is a linearly independent set, it is a basis for ( V ).

Finding the Basis

• To find the basis for ( V ) defined by ( V = {x | Ax = 0} ):
  • Create the augmented matrix [A|0],
  • Transform it into reduced row echelon form (rref) to identify non-zero rows, which provide the basis.