7 Linear Algebra IV

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Question 2 of the previous homework continued: (f) Find the rank of $A$. (g) Explain why the sum of the rank of $A$ and the dimension of the null space equals the number of columns of $A$. (h) What is the dimension of the row space of $A$? Find a fully simplified basis for it. Write the expression for the row space in terms of that basis. (i) Repeat the previous question for the column space. No, do not use $U$ here, that is wrong as the column space gets destroyed going from $A$ to $U$. (j) Neatly draw the two basis vectors of the column space in a three dimensional coordinate system and so illustrate a triangular piece of the column space plane.

2. Given
   
   $$A = \left( \begin{array}{rrrr} 4 & 3 & -5 & 6 \\ 1 & ... ... \\ 0 & -5 & 1 & 7 \\ 8 & 9 & 0 & 15 \end{array} \right)$$
   
   
   Find the determinant of this matrix using minors. Minimize the algebra in doing so. No Gaussian elimination steps, including partial pivoting, allowed.

3. Reconsider the matrix:
   
   $$A = \left( \begin{array}{rrrr} 4 & 3 & -5 & 6 \\ 1 & ... ... \\ 0 & -5 & 1 & 7 \\ 8 & 9 & 0 & 15 \end{array} \right)$$
   
   
   Find the determinant of this matrix using Gaussian elimination. Compare to your earlier result.

4. For the matrix of the previous two questions, find one row or column (state which row or column you took) of the inverse matrix using minors. Select the row or column so that you can reuse your earlier minors without recomputing.

5. For the matrix
   
   $$A = \left( \begin{array}{rrr} 11 & 0 & -5 \\ 0 & 1 & 0 \\ 4 & -7 & 9 \end{array} \right)$$
   
   
   find the inverse both using minors and using Gaussian elimination. Make sure the result is the same.

6. For the matrix
   
   $$A = \left( \begin{array}{rrr} 6 & -2 \\ -3 & 4 \end{array} \right)$$
   
   
   find the eigenvalues and eigenvectors. Take $\lambda_1$ to be the larger of the eigenvalues and $\lambda_2$ the smaller one. Is the matrix defective? Singular? What is the rank?