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.

Test your knowledge of key concepts in linear algebra with these flashcards from Purdue's MA 265 course. Topics include singular matrices, elementary matrices, and determinants. Perfect for quick review and exam preparation.