Linear Algebra Exam 1 Review

Matrix Equivalence and Solutions

• Every matrix is row equivalent to a unique matrix in reduced row-echelon form (RREF), which is distinct for each matrix.
• A system of n linear equations in n variables can have either unique, infinite, or no solutions, not limited to at most n solutions.
• If a system has two different solutions, it must have infinitely many due to the nature of linear equations.
• A system without free variables should still be examined carefully, as it may not guarantee a unique solution.

Augmented Matrices and Consistency

• Transformed augmented matrices [A b] and [C d] maintain equivalent solution sets when obtained through elementary row operations.
• If a system Ax=b has more than one solution, the corresponding homogeneous system Ax=0 also has or continues to have multiple solutions.
• With an mxn matrix A, if Ax=b is consistent for some b, it does not imply that A spans R^m.

Echelon Forms and Pivot Columns

• Consistency in the equation Ax=b is uncertain even if an augmented matrix can be transformed to reduced echelon form.
• Row equivalent matrices A and B have the same reduced echelon form.
• In an mxn matrix A, there must be m pivot columns if Ax=b is consistent for every b in R^m.
• A matrix A with a pivot position in every row ensures unique solutions for Ax=b for each b in R^m.

Linear Independence and Transformations

• An nxn matrix with n pivot positions does not necessarily reduce to the identity matrix.
• If 3x3 matrices A and B have pivot positions in all rows, they can be transformed into each other using elementary row operations.
• A matrix A can lead to multiple solutions in Ax=c if it already has multiple solutions for Ax=b.

Span and Linear Combinations

• For a set S={v1,v2,v3} in R^3, not being scalar multiples does not guarantee linear independence.
• A linearly independent set {u,v,w} cannot reside in R^2 but must be in R^3 or higher.
• The possibility of four vectors spanning R^m is not guaranteed.

Properties of Vector Transformations

• Any vector -u, derived from vector u in R^m, can be included in the span of {u,v}.
• Nonzero vectors u, v, and w in R^2 may not guarantee that w is a linear combination of u and v.
• In R^n, if w is a linear combination of u and v, it does not imply that u is expressible as a combination of v and w.

Linear Transformations and Mapping

• A linear transformation is fundamentally a function mapping vectors in one vector space to another.
• A linear transformation from a 6x5 matrix cannot cover R^6 completely due to dimensionality constraints.
• An mxn matrix A with m pivot columns will not ensure a one-to-one mapping unless each column also has a pivot, necessitating at least as many pivot columns as total columns.

Prepare for your Linear Algebra Exam 1 with these flashcards that cover key concepts and definitions. Each card presents a statement regarding linear equations and matrices, alongside whether the statement is true or false. Boost your understanding of row equivalency, solutions of equations, and echelon forms.