Questions and Answers
If the equation Ax=0 has only the trivial solution, then A is row equivalent to the n x n identity matrix.
True
If the columns of A span R^n, then the columns are linearly independent.
True
If A is an n x n matrix, then the equation Ax=b has at least one solution for each b in R^n.
False
If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.
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If the transpose of A is not invertible, then A is not invertible.
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If there is an n x n matrix D such that AD = I, then DA = I.
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If the linear transformation x|--> Ax maps R^n onto R^n, then the row reduced echelon form of A is I.
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If the columns of A are linearly independent, then the columns of A span R^n.
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If the equation Ax=b has at least one solution for each b in R^n, then the transformation x |--> Ax is not one-to-one.
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If there is a b in R^n such that the equation Ax=b is consistent, then the solution is unique.
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It is possible for a 4 x 4 matrix to be invertible when its columns do not span R^4.
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A square matrix with two identical columns can be invertible.
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If an n x n matrix G cannot be row reduced to I, what can be said about the columns of G?
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Study Notes
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.
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Description
Test your understanding of Linear Algebra concepts with this True/False quiz based on Chapter 2. Each question evaluates foundational ideas including solutions to equations and properties of matrices. Perfect for reinforcing key topics in the subject.
