Linear Algebra Chapter 2 True/False Quiz

Matrix Properties and Solutions

• If Ax=0 only has the trivial solution, A is row equivalent to the n x n identity matrix, ensuring a pivot in every row and column.
• Columns of A span R^n only if they are linearly independent, which occurs without free variables in a square matrix.

System of Equations

• An n x n matrix A ensures the equation Ax=b has at least one solution for each b in R^n only if A is invertible.
• A nontrivial solution in Ax=0 indicates A has fewer than n pivot positions, resulting in free variables and infinitely many solutions.

Invertibility Conditions

• Transpose A is not invertible if and only if A is not invertible.
• If there exists an n x n matrix D such that AD = I, then DA = I as well.

Linear Transformations

• If the linear transformation x |--> Ax maps R^n onto R^n, then the row reduced echelon form of A is the identity matrix I.
• Linear independence of columns in A guarantees that the columns span R^n.

One-to-One Transformations

• If Ax=b has at least one solution for every b in R^n, the transformation x |--> Ax can still be one-to-one.
• Having a consistent equation Ax=b does not imply a unique solution; there may be infinitely many solutions.

Matrix Invertibility and Linear Dependence

• A 4 x 4 matrix cannot be invertible if its columns do not span R^4; invertibility necessitates pivot positions in every column and row.
• A square matrix with two identical columns cannot be invertible due to linear dependency, which violates the condition for invertibility.

Linear Dependence and Spanning

• If an n x n matrix G cannot be row reduced to I, its columns are linearly dependent and do not span R^n.