Introduction to Linear Algebra 1 Flashcards

Linear Equations

• A linear equation can be expressed in the form: ( a_1x_1 + a_2x_2 + ... + a_nx_n = b ), where ( a_1, a_2, ..., a_n ) are real numbers.

Systems of Linear Equations

• A linear system is a collection of linear equations that share the same variables.
• A solution to a linear system is an ordered n-tuple of real numbers that satisfies each equation in the system.

Solving Linear Systems

• Finding the solution set of a linear system involves determining all possible solutions.
• Two linear systems are equivalent if they share the same solution set.

Intersection of Lines

• Solving a two-variable system equates to finding the intersection point of two lines.
• A linear system can have no solutions, exactly one solution, or infinitely many solutions.

Matrices

• A matrix is a rectangular arrangement of numbers organized into rows and columns.
• Elementary row operations include row replacement, row interchange, and row scaling.

Gauss Elimination and Row Operations

• The Gauss elimination method simplifies systems by keeping one variable and eliminating it from others, progressing through equations systematically.
• Row equivalent matrices can be transformed into one another via a series of row operations.

Echelon Forms

• A matrix is in Echelon form if all nonzero rows are above rows of zeros and if leading entries move rightward down rows.
• A matrix is in Reduced Echelon form (rref) if it fulfills all criteria of Echelon form plus additional conditions on leading entries.

Consistency and Solutions

• A linear system is consistent if the last column of its augmented matrix does not form a pivot column.
• A consistent linear system has a unique solution when free variables are absent.

Matrix Operations and Vector Spaces

• A column vector is a matrix with a single column.
• ( \mathbb{R}^n ) represents all vectors in n dimensions with real coefficients.

Linear Combinations and Spanning

• The span of vectors ( v_1, ..., v_p ) is the set of all possible linear combinations.
• A vector equation and its corresponding augmented matrix are equivalent if the resulting system is consistent.

Homogeneous Systems

• A system is homogeneous if ( b = 0 ) and always has the trivial solution ( x = 0 ).
• Nontrivial solutions exist in a homogeneous system if there are infinite solutions or free variables present.

Linear Dependence and Independence

• Vectors ( v_1, ..., v_p ) are linearly dependent if a nontrivial linear combination equates to zero.
• Linear independence occurs when the only solution to the equation involving the vectors is the trivial solution.

Transformations

• A transformation ( T:\mathbb{R}^n \rightarrow \mathbb{R}^m ) is linear if it satisfies vector addition and scalar multiplication properties.
• The augmented matrix transformation is a linear transformation.

Onto and One-to-One Mappings

• A transformation is onto if every possible output in ( \mathbb{R}^m ) can be achieved, meaning ( Ax = b ) is consistent.
• A transformation is one-to-one if the corresponding homogeneous equation has only the trivial solution and ensures no free variables exist.

Test your knowledge with these flashcards covering the basics of Linear Algebra. Each card presents key concepts including linear equations and systems of linear equations. Perfect for students looking to reinforce their understanding of introductory linear algebra topics.