Let’s analyze each of these statements and clarify whether they are true or false:

1. If the equation $A x = 0$ has a nontrivial solution, then $A$ has fewer than $n$ pivots.

   True: If $A x = 0$ has a nontrivial solution, then the matrix $A$ has at least one free variable, meaning it is not full-rank. This implies that $A$ has fewer than $n$ pivots (where $n$ is the number of columns of $A$).

2. If $A$ is not invertible, then $A^2$ is also not invertible.

   True: If $A$ is not invertible, then $A$ has a determinant of zero. The matrix $A^2 = A \times A$ will also have a determinant of zero, making $A^2$ not invertible as well.

3. If the linear transformation $T_A$ is one-to-one, then the columns of $A$ form a linearly dependent set.

   False: If the transformation $T_A$ is one-to-one, it implies that the equation $A x = 0$ has only the trivial solution. This means the columns of $A$ are linearly independent, not dependent.

4. If the linear transformation $T_A$ is onto, then it is also one-to-one.

   False: A linear transformation can be onto (surjective) without being one-to-one (injective). Surjectivity implies that every element in the codomain is mapped by at least one element in the domain, but it doesn't necessarily mean that each element in the domain maps to a unique element.

5. If the equation $A x = 0$ has the trivial solution, then the columns of $A$ span $\mathbb{R}^n$.

   False: The equation $A x = 0$ having the trivial solution (i.e., $x = 0$ being the only solution) means that the columns are linearly independent, but this does not necessarily imply that they span $\mathbb{R}^n$. The matrix must be square (and invertible) for the columns to span $\mathbb{R}^n$.

6. If $A$ is not invertible, then $A^7$ is also not invertible.

   True: If $A$ is not invertible, then $A$ has a determinant of zero. Any power of $A$, including $A^7$, will still have a determinant of zero, making it non-invertible.

Would you like more details or have any specific questions?

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Here are 5 related questions you can explore:

1. What are the conditions for a matrix to be invertible?
2. How does the rank of a matrix relate to the number of pivots it has?
3. What is the difference between a transformation being one-to-one and onto?
4. How does the determinant of a matrix behave under matrix multiplication and exponentiation?
5. What is the relationship between linear dependence and the spanning of a vector space?

Tip: If a matrix has linearly independent columns, it is automatically one-to-one and invertible.