Your assignment includes problems on linear algebra topics such as basis verification, coordinate representation, reduced row-echelon form (RREF), basis finding, and determinant calculation. Here’s a brief outline of how you can approach each problem:

Problem 1: Basis and Coordinate Representation

1. Bases Verification (Part i):

   • Show that $B = \{u_1, u_2, u_3\}$ and $B' = \{v_1, v_2, v_3\}$ are linearly independent.
   • Confirm they span $\mathbb{R}^3$.

2. Coordinate Representations (Part ii):

   • Solve for $[v]_B$ and $[v]_{B'}$:
     • Set up $v = c_1u_1 + c_2u_2 + c_3u_3$ for $[v]_B$ and similarly for $B'$.
     • Solve the systems of equations for $c_1, c_2, c_3$.

3. Change of Basis Matrix (Part iii):

   • Calculate $A_{B \leftarrow B'} = P^{-1}$ where $P$ is the matrix with $B'$'s vectors as columns in $B$-coordinates.
   • Verify that $A_{B \leftarrow B'} \cdot A_{B' \leftarrow B} = I$.

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Problem 2: Matrix Properties

1. RREF (Part i):

   • Use Gaussian elimination to find the reduced row-echelon form of $A$.

2. Basis (Part ii):

   • For $\text{row}(A)$: Extract non-zero rows of RREF.
   • For $\text{col}(A)$: Use pivot columns from RREF in the original matrix.
   • For $\text{null}(A)$: Solve $A\mathbf{x} = 0$.

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Problem 3: Determinant Calculation

1. Determinant of 3x3 Matrix (Part i):

   • Use cofactor expansion to compute $\text{det}(A)$.

2. Determinant with Row Operations (Part ii):

   • Apply the given row operations step-by-step.
   • Adjust for any multiplication of rows affecting the determinant:
     • Row operation effects: $R_k \to R_k + cR_j$ leaves the determinant unchanged.
     • Verify $\text{det}(A)$ after operations.

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Would you like a detailed walkthrough of any specific question or all of them? Let me know if you'd prefer detailed calculations or conceptual clarifications.

Here are five questions that expand on the information:

1. What conditions are necessary for a set of vectors to form a basis in $\mathbb{R}^n$?
2. How do row operations affect the determinant of a matrix?
3. Can a matrix with linearly dependent columns have an invertible change of basis matrix?
4. How does the rank of a matrix relate to the dimensions of its null space and column space?
5. What is the significance of the reduced row-echelon form in solving linear systems?

Tip: When solving linear systems for coordinate representations, writing augmented matrices and solving systematically reduces computation errors.