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Linear Algebra Test 3 Topics

Author

Chalmeta

Concepts from Chapters 5 and 6

  1. Identify whether a set of vectors forms a basis for a set.
  2. Know that the dimension of a vector space is equal to the number of vectors in its basis.
  3. Know what the rank and nullity of a matrix are.
  4. What eigenvalues and eigenvectors represent. Solutions to \(Ax = \lambda x\).
  5. How to calculate the eigenvalues of a small matrix (\(2 \times 2\) or \(3 \times 3\)) by hand.
  6. How to calculate the eigenvectors of a matrix.
  7. What the algebraic multiplicity of an eigenvalue is.
  8. Chapter 5, Theorem 2: If \(\{v_1, v_2, \ldots, v_r\}\) are eigenvectors which correspond to distinct eigenvalues, then the set \(\{v_1, v_2, \ldots, v_r\}\) is linearly independent.
  9. How to write the characteristic equation of a matrix and what it is used for.
  10. How to diagonalize a matrix. When would diagonalization be advantageous?
  11. How to add, subtract, and multiply complex numbers.
  12. How to find the complex eigenvalues and eigenvectors of a matrix.
  13. Compute the dot product of a pair of vectors.
  14. Compute the norm of a vector.
  15. Determine the angle between a pair of vectors.
  16. Identify orthogonal vectors and sets.
  17. Find the projection of a vector onto a subspace.
  18. Find the shortest distance from a vector to a subspace.
  19. Given an orthogonal basis \(\{u_1, u_2, \ldots, u_r\}\) for a subspace \(W\), write vector \(y \in W\) as \(y = c_1 u_1 + c_2 u_2 + \cdots + c_r u_r\).
  20. Use the Gram-Schmidt process to construct an orthonormal basis for subspace \(W\).

Practice Problems

Problem 1. Identify whether the following statements are true or false. If a statement is false, give an explanation why.

  1. An \(n \times n\) matrix has \(n\) eigenvalues.
  2. \(\dfrac{6 - 12i}{2 + 3i} = 3 - 4i\)
  3. If \(A\) is similar to the identity matrix, then \(\det A = 0\).
  4. If \(x\) is an eigenvector of \(A\), then the line through the origin and \(x\) passes through \(Ax\).
  5. If \(B\) and \(C\) are bases for the same vector space \(V\), then \(B\) and \(C\) contain the same number of vectors.
  6. If \(u\) and \(v\) are in \(\mathbb{R}^n\), then \(u \cdot v = v \cdot u\).
  7. \(A\) is a diagonalizable matrix if \(A = PDP^{-1}\) for some diagonal matrix \(D\) and some invertible matrix \(P\).

Problem 2. You are given that \[x_1 = \begin{bmatrix}1\\1\\0\end{bmatrix}, \quad x_2 = \begin{bmatrix}1\\0\\2\end{bmatrix}, \quad x_3 = \begin{bmatrix}0\\1\\-1\end{bmatrix}, \quad P = \begin{bmatrix}x_1 & x_2 & x_3\end{bmatrix}, \quad D = \begin{bmatrix}2&0&0\\0&5&0\\0&0&-4\end{bmatrix}\] and \(A = PDP^{-1}\). Then \(A^2 x_2 =\)

  1. \(\begin{bmatrix}25\\0\\50\end{bmatrix}\)
  2. \(\begin{bmatrix}25\\0\\100\end{bmatrix}\)
  3. \(\begin{bmatrix}2\\0\\-8\end{bmatrix}\)
  4. \(\begin{bmatrix}0\\0\\0\end{bmatrix}\)
  5. \(\begin{bmatrix}5\\0\\20\end{bmatrix}\)

Problem 3. Find the eigenvalues and eigenvectors of \(A = \begin{bmatrix}-4 & 6 \\ -1 & 1\end{bmatrix}\).

Problem 4. Find the eigenvalues and eigenvectors of \(A = \begin{bmatrix}-3 & 4 \\ 3 & 8\end{bmatrix}\).

Problem 5. Find the eigenvalues and eigenvectors of \(A = \begin{bmatrix}3 & -5 \\ 9 & 3\end{bmatrix}\).

Problem 6. Let \(A = \begin{bmatrix}4 & 3 & 2 \\ 2 & 9 & 4 \\ -1 & -3 & 1\end{bmatrix}\). The eigenvalues of \(A\) are \(\lambda_1 = 3\) and \(\lambda_2 = 8\). Find an eigenbasis for each eigenvalue.

Problem 7. The \(3 \times 3\) matrix \(A\) has eigenvalues \(\lambda_1 = 3\), \(\lambda_2 = -4\), \(\lambda_3 = 1\), with corresponding eigenvectors \[v_1 = \begin{bmatrix}1\\1\\3\end{bmatrix}, \quad v_2 = \begin{bmatrix}0\\1\\1\end{bmatrix}, \quad v_3 = \begin{bmatrix}0\\0\\1\end{bmatrix}.\] Write \(A\) in the form \(PDP^{-1}\), where \(D\) is a diagonal matrix.

Problem 8. Let \(A = \begin{bmatrix}1&0&1\\0&2&2\\0&1&1\end{bmatrix}\). Which of the following vectors is orthogonal to the column space of \(A\)?

  1. \(\begin{bmatrix}1\\1\\-1\end{bmatrix}\)
  2. \(\begin{bmatrix}1\\4\\2\end{bmatrix}\)
  3. \(\begin{bmatrix}0\\1\\-2\end{bmatrix}\)
  4. \(\begin{bmatrix}2\\0\\2\end{bmatrix}\)
  5. Since the second row is a multiple of the third row, Col \(A\) is undefined.

Problem 9. Let \(A = \begin{bmatrix}4&2&-3\\3&4&1\\4&1&5\end{bmatrix}\). Then \(\lambda = 3\) is an eigenvalue corresponding to the eigenvector \(v = \begin{bmatrix}1\\-2\\a\end{bmatrix}\). Find the value of \(a\).

Problem 10. Let \[A = \begin{bmatrix}5&-1&3&-1\\0&4&h&0\\0&0&5&4\\0&0&0&1\end{bmatrix}.\] Find \(h\) so that the eigenspace corresponding to the eigenvalue \(\lambda = 5\) is 2-dimensional.

Problem 11. Suppose the \(2 \times 2\) matrix \(A\) has eigenvalues \(\lambda_1 = 4\) and \(\lambda_2 = 3\) with eigenvectors \(v_1\) and \(v_2\), respectively. If \(u = 5v_1 + v_2\), then \(A^2 u\) is equal to:

  1. \(25v_1 + v_2\)
  2. \(25v_1 + 3v_2\)
  3. \(80v_1 + 9v_2\)
  4. \(100v_1 + 3v_2\)
  5. \(400v_1 + 9v_2\)

Problem 12. An \(n \times n\) matrix \(B\) has characteristic polynomial \(p(\lambda) = -\lambda(\lambda - 3)^3(\lambda - 2)^2(\lambda + 1)\). Which of the following statements is FALSE?

  1. rank \(B = 6\).
  2. \(\det(B) = 0\).
  3. \(\det(B^T B) = 0\).
  4. \(B\) is invertible.
  5. \(n = 7\).

Problem 13. Given vectors \(u = \begin{bmatrix}10\\0\\5\end{bmatrix}\) and \(v = \begin{bmatrix}2\\3\\-1\end{bmatrix}\), find \(u \cdot v\).

Problem 14. Find a unit vector in the direction of \(v = \begin{bmatrix}2\\3\\-1\end{bmatrix}\).

Problem 15. Find the shortest distance between the two vectors \(u = (6, -12)\) and \(v = (-12, 12)\).

Problem 16. Determine whether the following set of vectors is orthogonal: \[\begin{bmatrix}-2\\-4\\-2\end{bmatrix}, \quad \begin{bmatrix}20\\0\\-20\end{bmatrix}, \quad \begin{bmatrix}-20\\-20\\-20\end{bmatrix}\]

Problem 17. Find the orthogonal projection of \(y = \begin{bmatrix}-24\\10\end{bmatrix}\) onto \(u = \begin{bmatrix}4\\20\end{bmatrix}\).

Problem 18. Let \(W\) be the subspace spanned by \(u_1 = \begin{bmatrix}1\\0\\-1\end{bmatrix}\) and \(u_2 = \begin{bmatrix}2\\1\\2\end{bmatrix}\). Write \(y = \begin{bmatrix}19\\3\\11\end{bmatrix}\) as the sum of a vector in \(W\) and a vector orthogonal to \(W\). Find the closest point to \(y\) in the subspace \(W\) and the shortest distance.

Problem 19. Let \(A = \begin{bmatrix}0&0&0\\1&0&0\\4&0&0\end{bmatrix}\). The number of linearly independent eigenvectors for the eigenvalue \(\lambda = 0\) is __________.

Problem 20. Suppose \(A = \begin{bmatrix}1&-1&-1&-1\\-1&1&-1&-1\\-1&-1&1&-1\\-1&-1&-1&1\end{bmatrix}\) and the only distinct eigenvalues are \(r = 2\) and \(r = -2\).

  1. Find a basis for the eigenspace associated with \(r = 2\).
  2. Complete: \(r = 2\) must be an eigenvalue of multiplicity __________ because:

Problem 21.

  1. In trying to find eigenvector(s) associated with an eigenvalue \(r = 2\), the reduced augmented matrix for \((A - 2I)x = 0\) is shown below. Find a basis for the eigenspace. \[\begin{bmatrix}1 & 0 & 3 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{bmatrix}\]

  2. Which of the following could NOT possibly be true? Circle all that apply:

For the matrix \(A\), 2 is an eigenvalue of multiplicity    1    2    3    4    5

Problem 22. Use the Gram-Schmidt process to find an orthonormal basis for Col \(A\) when \[A = \begin{bmatrix}1 & 2 & 19 \\ 0 & 1 & 3 \\ -1 & 2 & 11\end{bmatrix}\]

Answers

Problem 1.

  1. True, including multiplicity.
  2. False.
  3. False — \(A = PIP^{-1} = I\), so \(\det A = 1\).
  4. True.
  5. True.
  6. True.
  7. True.

Problem 2. (a) \(\begin{bmatrix}25\\0\\50\end{bmatrix}\)

Problem 3. \[E_{\lambda = -2} = \left\{ \begin{bmatrix}3\\1\end{bmatrix} \right\}, \qquad E_{\lambda = -1} = \left\{ \begin{bmatrix}2\\1\end{bmatrix} \right\}\]

Problem 4. \[E_{\lambda = 9} = \left\{ \begin{bmatrix}1\\3\end{bmatrix} \right\}, \qquad E_{\lambda = -4} = \left\{ \begin{bmatrix}-4\\1\end{bmatrix} \right\}\]

Problem 5. \[E_{\lambda = 3 + 3i\sqrt{5}} = \left\{ \begin{bmatrix}i\sqrt{5}\\3\end{bmatrix} \right\}\]

Problem 6. \[E_{\lambda = 3} = \left\{ \begin{bmatrix}-2\\0\\1\end{bmatrix},\, \begin{bmatrix}-3\\1\\0\end{bmatrix} \right\}, \qquad E_{\lambda = 8} = \left\{ \begin{bmatrix}-1\\-2\\1\end{bmatrix} \right\}\]

Problem 7. \[A = \begin{bmatrix}1&0&0\\1&1&0\\3&1&1\end{bmatrix} \begin{bmatrix}3&0&0\\0&-4&0\\0&0&1\end{bmatrix} \begin{bmatrix}1&0&0\\-1&1&0\\-2&-1&1\end{bmatrix}\]

Problem 8. (c) \(\begin{bmatrix}0\\1\\-2\end{bmatrix}\)

Problem 9. \(a = -1\)

Problem 10. \(h = 3\)

Problem 11. (c) \(80v_1 + 9v_2\)

Problem 12. (d) \(B\) is invertible is the false statement, because \(\lambda = 0\) is a root of the characteristic polynomial, meaning \(\det B = 0\), so \(B\) is singular.

Problem 13. \(u \cdot v = 15\)

Problem 14. \[\hat{v} = \begin{bmatrix}2/\sqrt{14}\\3/\sqrt{14}\\-1/\sqrt{14}\end{bmatrix}\]

Problem 15. Distance for projection of \(u\) onto \(v\) is \(3\sqrt{2}\). Distance for projection of \(v\) onto \(u\) is \(\dfrac{12\sqrt{5}}{5}\) (shortest).

Problem 16. No — the set is not orthogonal.

Problem 17. \[\text{proj}_u\, y = \begin{bmatrix}1\\5\end{bmatrix}\]

Problem 18. \[y = \begin{bmatrix}18\\7\\10\end{bmatrix} + \begin{bmatrix}1\\-4\\1\end{bmatrix}\] Closest point: \((18, 7, 10)\). Shortest distance: \(\sqrt{18}\).

Problem 19. 2

Problem 20.

  1. \(E_{r=2} = \left\{ \begin{bmatrix}-1\\1\\0\\0\end{bmatrix},\, \begin{bmatrix}-1\\0\\1\\0\end{bmatrix},\, \begin{bmatrix}-1\\0\\0\\1\end{bmatrix} \right\}\)

  2. \(r = 2\) must be an eigenvalue of multiplicity 3.

Problem 21.

  1. \(E_{r=2} = \left\{ \begin{bmatrix}0\\1\\0\\0\end{bmatrix},\, \begin{bmatrix}-3\\0\\1\\0\end{bmatrix},\, \begin{bmatrix}-1\\0\\0\\1\end{bmatrix} \right\}\)

  2. Could NOT possibly be true: 1, 2, 5 — because the number of eigenvectors found is 3, the multiplicity is at least 3 (a lower bound on multiplicity), and there are at most 4 eigenvectors total since it is a \(4 \times 4\) matrix, so multiplicity cannot be 5

Problem 22. \[\left\{ \begin{bmatrix}1/\sqrt{2}\\0\\-1/\sqrt{2}\end{bmatrix},\; \begin{bmatrix}2/3\\1/3\\2/3\end{bmatrix},\; \begin{bmatrix}1/(3\sqrt{2})\\-4/(3\sqrt{2})\\1/(3\sqrt{2})\end{bmatrix} \right\}\]