Exercises

Vector spaces:
  1. Let V be a complex inner product space and let \( {\bf u}, {\bf v} \in V . \) Show that
    \[ \left\langle {\bf u} , {\bf v} \right\rangle = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{{\bf j} \theta} \left\| {\bf u} + e^{{\bf j} \theta} {\bf v} \right\|^2 {\text d}\theta . \]
  2. For the following problems, use Gauss-Jordan Elimination to put the matrices in RREF. \[ (a) \quad A= \begin{bmatrix} 2 & -2 & 4 & -2\\ 2 & 1 & 10 & 7\\ -4 & 4 & -8 & 4\\ 4 & -1 & 14 & 6 \end{bmatrix}, \qquad\quad (b) \quad A= \begin{bmatrix} 1 & 3 & 4 & 14\\ 2 & -3 & 2 & 10\\ 3 & -1 & 1 & 9\\ \end{bmatrix} \]
  3. If matrix A represents an augmented matrix system of equations of the form \( ax_{1}+bx_{2}+cx_{3}=d, \) find the values of x1, x2, and x3. \[ (a) \quad A= \begin{bmatrix} 1 & 2 & 2 & 1\\ 2 & 3 & 6 & 0 \\ 3 & 4 & 7 & 0\\ \end{bmatrix}, \qquad\quad (b) \quad A= \begin{bmatrix} 2 & 4 & -1 & -9\\ -1 & 1 & -1 & -6 \\ -3 & 3 & 4 & 3\\ \end{bmatrix} \]
Gaussian Elimination
  1. For which two numbers a will elimination fail on matrix \( \begin{bmatrix} a&1 \\ a&1 \end{bmatrix} ? \)
  2. For which hree numbers a will elimination fail to give three pivots? \( \begin{bmatrix} a&1&4 \\ a&a&3 \\ a&a&a \end{bmatrix} . \)
Matrix transformations:
  1. Find the matrix for the transformation. \[ \mbox{(a)} \quad S \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} \right) = \begin{bmatrix} 1x_{1}+4x_{2}-5x_{3}\\ -3x_{1}+3x_{2}-2x_{3}\\ 1x_{1}+-2x_{2}+6x_{3} \end{bmatrix}; \qquad \mbox{(b)} \quad T \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} \right) = \begin{bmatrix} 5x_{1}+3x_{2}-4x_{3}\\ x_{2}+2x_{3}\\ \end{bmatrix}; \] \[ \mbox{(c)} \quad U \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} \right) = \begin{bmatrix} x_{1}-x_{2}+x_{3}\\ 3x_{1}+x_{2}-x_{3}\\ 2x_{1}+2x_{2}+x_{3} \end{bmatrix} \]
  2. Is the following transformation linear? Check that both axioms T(A+B)=T(A)+T(B) and T(c*A)=c*T(A) are satisfied. \[ \mbox{(a)} \quad S \left( \begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix} \right) = \begin{bmatrix} -x_{1}+3x_{2}\\ 0\\ \end{bmatrix}; \qquad \mbox{(b)} \quad T \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} \right) = \begin{bmatrix} 2x_{1}-4\\ x_{2}-2x_{3}\\ \end{bmatrix}; \] \[ \mbox{(c)} \quad U \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ \end{bmatrix} \right) = \begin{bmatrix} x_{1}-x_{2}\\ 3x_{2}+2x_{3}\\ -x_{3}-4x_{4}\\ \end{bmatrix}; \qquad \mbox{(d)} \quad V \left( \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ \end{bmatrix} \right) = \begin{bmatrix} x_{1}+3x_{2}+2x_{3}+1x_{5}\\ -x_{1}-x_{2}-x_{3}+x_{4}+x_{6}\\ 4x_{2}+2x_{3}+4x_{4}+3x_{5}+3x_{6}\\ x_{1}+3x_{2}+2x_{3}-2x_{4}\\ \end{bmatrix} . \]
  3. Find the composition of transformations S with T, i.e. S∘T.
    1. \[ \quad S:\ \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} \,\mapsto \, \begin{bmatrix} x_{1}+3x_{2} \\ 6x_{1}+5x_{2} \\ 3x_{1}+8x_{2} \end{bmatrix} \quad T:\ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix}\,\mapsto \, \begin{bmatrix} x_{1}+3x_{2}+7x_{3} \\ -x_{1}-2x_{2}+6x_{3} \\ -4x_{1}+x_{2}-7x_{3} \\ \end{bmatrix} \]
    2. \[ \quad S:\ \begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix} \,\mapsto \, \begin{bmatrix} -2x_{1}+3x_{2} \\ 4x_{2} \\ -3x_{1}-2x_{2} \\ x_{1}-2x_{2} \end{bmatrix}, \quad T :\ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{bmatrix}\,\mapsto \, \begin{bmatrix} x_{2}-2x_{3}+x_{4} \\ 2x_{1}+5x_{2}-2x_{3} \\ -x_{1}+2x_{2}-3x_{3}+2x_{4} \end{bmatrix} . \]

Lebel the following statements as being true or false.
  1. Another notation for the vector \( \begin{bmatrix} 3 \\ -4 \end{bmatrix} \) is [ 3 -4 ].
  2. The set span{ u, v } of two vectors u and v is always visualized as a plane through the origin.
  3. An example of a linear combination of vectors u and v is 2u.
  4. For two nonzero vectors u and v from ℝn, span{ u, v } contains the line through v and the origin.
  5. The set span{ u, v } is always visualized as a plane through the origin.
  6. If T : V → U is a linear transformation from one vector space into another vector space, then T carries linearly independent subsets of V onto linearly independent subsets of U.
  7. If T : V ↦ U is a linear transformation from one vector space into another vector space, then T preserves scalar products.
  8. Is the following transformation $\DS S \,:\, \mathbb{C}^3 \,\to\,\mathbb{C}^3$ linear? \[ S \left( \begin{bmatrix} x_1 \\ 2\,x_2 \\ x_3 \end{bmatrix} \right) = \begin{bmatrix} 3\,x_1 - x_3 \\ -x_1 + 2\,x_2 \\ x_2 + 3\,x_3 -5 \end{bmatrix} , \]
  9. Any system of n linear equations in n variables has at most n solutions.
  10. Every matrix is row equivalent to a unique matrix in echelon form.
  11. If a system of linear equations has two distinct solutions, it must have infinitely many solutions.
  12. If an augmented matrix [A b] is transformed into [B c] by elementary row operations, then the equations Ax = b and Bx = c have exactly the same solution set.
  13. If u and v are in ℝm, then -u is in Span{ u, v }.
  14. If a system Ax = b has more than one solution, then so does the system Ax = 0.
  15. If A is an m×n matrix and the equation Ax = b is consistent for some b, then the columns of A span ℝm.
  16. If none of the vectors in the set S= { u1, u2, u3 } in ℝ³ is a multiple of one of the other vectors, then S is linearly independent.
  17. If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.
  18. Any set containing the zero vector is linearly dependent.
  19. Subsets of a linearly dependent set are linearly dependent.
  20. If w is a linear combination of u and v in ℝm, then u is a linear combination of v and w.
  21. If S is a linearly dependent set, then each element of S is a linear combination of other elements of S.
  22. If A is an m×n matrix and the equation Ax = b is consistent for every b in ℝm, then A has m pivot columns.
  23. If an m×n matrix A has a pivot position in every row, then the equation Ax = b has aunique solution for each b in ℝm.
  24. If an m×n matrix A has n pivot positions, then the reduced echelon form of A is the n×n identity matrix.
  25. The empty set is linearly dependent.
  26. Subsets of linearly independent sets are linearly independent.
  27. The span of ∅ is ∅.
  28. The zero vector is a linear combination of any nonempty set of vectors.
  29. In solving a system of linear equations it is permissible to add a multiple of one equation to another.
  30. Every system of linear equations has a solution.
  31. If 3×3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.
  32. Any system of n linear equations in n variables has at most n solutions.
  33. For a m×n matrix A, if the equation Ax = b has at least two different solutions, and if the equation Ax = c is consistent, then the equation Ax = c has many solutions.
  34. If m×n matrices A and B are row equivalent and if the columns of A span ℝm, then so do the columns of B.
  35. If none of the vectors in the set S = { v1, v2, v3 } in ℝ³ is a multiple of one of the other vectors, then S is linearly independent.
  36. If u, v, and w are nonzero vectors in ℝ², then u is a linear combination of v and w.
  37. If A is an m×n matrix with m pivot columns, then the linear transformation x ↦ Ax is a one-to-one mapping.