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Linear Algebra Test 2 Topics

Author

Chalmeta

Concepts from Chapters 2, 3, and 4

  1. What is a determinant?
  2. Know how to use Cramer’s rule.
  3. Know what a cofactor is and how to use them to compute determinants.
  4. How elementary row operations affect the determinant of a matrix (Theorem 3, p. 171).
  5. How to combine row operations and cofactor expansion efficiently.
  6. The connection between determinants and invertibility.
  7. How to compute the area of a triangle (and other polygons) using determinants.
  8. The equivalent conditions of the Invertible Matrix Theorem. (a) – (r)
  9. The definition of a vector space.
  10. The three criteria one has to check to see if a subset of \(\mathbb{R}^n\) is a subspace.
  11. \(\text{Span}\{v_1, v_2, \ldots, v_p\}\) is the set of all linear combinations \(c_1 v_1 + c_2 v_2 + \cdots + c_p v_p\).
  12. The span of a set of vectors in \(\mathbb{R}^n\) is always a subspace of \(\mathbb{R}^n\).
  13. Compute the null space of a matrix and express that null space as the span of a set of vectors.
  14. Determine whether a vector is in the null space of a given matrix.
  15. Compute the column space of a matrix and express that column space as the span of a set of vectors.
  16. Understand that if \(A\) is the standard matrix of a linear transformation, Nul \(A\) is a subset of the domain, and Col \(A\) is the range.
  17. Determine whether a set of vectors forms a basis for its span.
  18. Compute a basis for Nul \(A\) and for Col \(A\).
  19. Know that the dimension of a vector space is equal to the number of vectors in its basis.
  20. Know what the rank and nullity of a matrix are.
  21. Use the rank of the matrix to answer questions like those in the right-hand column of page 248.
  22. Given a basis \(\mathcal{B}\) and a vector \(x\), find \([x]_\mathcal{B}\).
  23. Given a basis \(\mathcal{B}\) and a vector \([x]_\mathcal{B}\), find \(x\).
  24. Given a matrix \(A\), construct a basis for Row \(A\) (= Col \(A^T\)) and find its dimension.

Practice Problems

Problem 1. Calculate the area of the parallelogram formed between the points \((0,3)\), \((2,4)\), \((5,2)\), and \((3,1)\).

Problem 2. The vector \(\begin{bmatrix} a \\ b \\ 10 \\ 5 \end{bmatrix}\) is in the null space of \(\begin{bmatrix} 2 & 3 & 0 & 1 \\ 1 & 4 & 1 & 2 \end{bmatrix}\). Find the values of \(a\) and \(b\).

Problem 3.

  1. Calculate the determinant of \[B = \begin{bmatrix} 0 & 4 & 0 & 1 & 0 \\ 9 & 1 & 0 & 11 & 3 \\ 3 & 6 & 0 & 8 & 0 \\ 2 & -5 & 4 & 1 & -7 \\ 0 & 2 & 0 & 1 & 0 \end{bmatrix}\] by using cofactor expansion efficiently.

  2. Now that you have the determinant of \(B\), what can you say about the determinant of the matrix \(C\) shown here: \[C = \begin{bmatrix} 9 & 1 & 0 & 11 & 3 \\ 0 & 12 & 0 & 3 & 0 \\ 3 & 6 & 0 & 8 & 0 \\ 2 & -5 & 4 & 1 & -7 \\ 0 & 2 & 0 & 1 & 0 \end{bmatrix}\]

Problem 4. Suppose you knew that the columns of the \(5 \times 5\) matrix \(A\) were linearly dependent. What can you say about the determinant of \(A\)?

Problem 5. Show that the set of vectors \(\begin{bmatrix} 2r + 3s \\ r - s \\ 5r \end{bmatrix}\) forms a subspace of \(\mathbb{R}^3\).

Problem 6. Show that the integer lattice \(\mathbb{Z}^2\), which is the set of all vectors \(\begin{bmatrix} x \\ y \end{bmatrix}\) where \(x\) and \(y\) are whole integers, does not form a subspace of \(\mathbb{R}^2\).

Problem 7. Suppose a \(4 \times 6\) matrix \(A\) has rank 2. Then:

  1. Nul \(A\) is a ______-dimensional subspace of \(\mathbb{R}^{\underline{\phantom{n}}}\).
  2. Col \(A\) is a ______-dimensional subspace of \(\mathbb{R}^{\underline{\phantom{n}}}\).

Problem 8.

  1. What is the maximum rank of a \(3 \times 7\) matrix?
  2. The \(4 \times 9\) matrix \(A\) has a rank of 3. What is its nullity?

Problem 9. If \(A\) is a \(9 \times 6\) matrix with rank \(A = 6\), what is the nullity of \(A\)?

Problem 10. If \(A\) is a \(4 \times 5\) matrix that is row equivalent to \[B = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix},\] what is the number of pivot positions that \(A\) has? What is the rank of \(A\)? What is the nullity of \(A\)? Can you name a basis for the row space of \(A\)? Why might the first, fourth, and fifth columns of \(B\) fail to form a basis for the column space of \(A\)?

Problem 11. If \(A\) is the matrix of a linear transformation from \(\mathbb{R}^3\) to \(\mathbb{R}^7\), and \(A\) has exactly three vectors in the basis of its null space, what is the dimension of the row space of \(A\)?

Problem 12. If \(A\) is a \(6 \times 3\) matrix, can \(A\) have a 4-dimensional row space? Can \(A\) have a 4-dimensional null space?

Problem 13. Let \(\mathcal{B} = \left\{ \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\}\), so \(\mathcal{B}\) is a basis for \(\mathbb{R}^2\). Express the vector \(x = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\) as a coordinate vector relative to \(\mathcal{B}\) (that is, find \([x]_\mathcal{B} = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}\)).

Problem 14. Let \(\mathcal{B} = \left\{ \begin{bmatrix} -1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\}\), so \(\mathcal{B}\) is a basis for \(\mathbb{R}^2\). Given the coordinates of \(x\) in this basis: \(c_1 = 3,~ c_2 = -3\), what are the coordinates of \(x\) in the standard basis?

Problem 15. Let \(A\) be an \(n \times n\) invertible matrix. Label the following as true or false:

  1. \(\dim \text{col}\, A^T = n\)
  2. If \(A \sim B\), then \(B\) is invertible.
  3. The rows of \(A\) are linearly independent and span \(\mathbb{R}^n\).
  4. The columns of \(A^T\) are linearly independent.
  5. \(\det A^T = -\det A\)
  6. If \(B\) contains exactly the same rows as \(A\), but in a different order, then \(B\) is invertible.
  7. The transformation \(T(x) = Ax\) is both one-to-one and onto.
  8. The equation \(Ax = 0\) has an infinite number of solutions.
  9. The reduced echelon form of \(A\) is an identity matrix.
  10. Nul \(A\) is a single point.

Problem 16. If the row space of \(A\) is a two-dimensional subspace of \(\mathbb{R}^3\), is it possible to determine the number of rows of \(A\)? How about the number of linearly independent rows of \(A\)?

Problem 17. If \(A\) is an \(n \times n\) matrix and \(A \sim I\), then do we know the rank of \(A^T\)?

Problem 18. Use a combination of row reduction and cofactor expansion to calculate \(\det B\) where \[B = \begin{bmatrix} -1 & 4 & 6 & 0 \\ 4 & 2 & 3 & 0 \\ 6 & 6 & 8 & 6 \\ 5 & 3 & 5 & 3 \end{bmatrix}\]

Problem 19. If \(A = \begin{bmatrix} 1 & 0 & -2 & -2 \\ 0 & 1 & 1 & 4 \\ 3 & -1 & -7 & 3 \end{bmatrix}\), find a basis for Nul \(A\) and Col \(A\).

Problem 20. Mark each statement TRUE or FALSE and explain why.

  1. If \(A\) is a \(2 \times 2\) matrix and \(\det A = 0\), then one column of \(A\) is a multiple of the other.
  2. If \(A\) is a \(3 \times 3\) matrix, then \(\det 5A = 5 \det A\).
  3. \(\det A^T A \geq 0\).
  4. A plane in \(\mathbb{R}^3\) is a two-dimensional subspace of \(\mathbb{R}^3\).
  5. If \(\{v_1, \ldots, v_n\}\) are vectors in a vector space \(V\), then \(\text{Span}\{v_1, \ldots, v_n\}\) is a subspace of \(V\).
  6. The set of pivot columns of a matrix is linearly independent.
  7. If \(A\) is a \(3 \times 5\) matrix, then Nul \(A\) is a subspace of \(\mathbb{R}^5\).
  8. If \(\mathcal{B}\) and \(\mathcal{C}\) are bases for the same vector space \(V\), then \(\mathcal{B}\) and \(\mathcal{C}\) contain the same number of vectors.
  9. If \(A\) is a \(3 \times 9\) matrix in echelon form, then rank \(A = 3\).

Problem 21. Use Cramer’s rule to solve the system \[\begin{aligned} 2x_1 - 3x_2 &= 7 \\ -3x_1 + 4x_2 &= 9 \end{aligned}\]

Answers

Problem 1. \[\det \begin{bmatrix} 2 & 1 \\ 3 & -2 \end{bmatrix} = -7, \quad \text{so the area is } |-7| = 7\]

Problem 2. \(a = 8\) and \(b = -7\).

Problem 3.

  1. \(\det B = 72\)

  2. \(\det C = -216\). Switch two rows (introduces a negative) and multiply one row by 3 (triples the determinant), giving \(-3 \times 72 = -216\).

Problem 4. \(\det A = 0\).

Problem 5. \[\begin{bmatrix} 2r + 3s \\ r - s \\ 5r \end{bmatrix} = \text{Span}\left\{ \begin{bmatrix} 2 \\ 1 \\ 5 \end{bmatrix}, \begin{bmatrix} 3 \\ -1 \\ 0 \end{bmatrix} \right\}\] All spans are subspaces. The set also satisfies closure under addition and scalar multiplication, contains the zero vector, and inherits standard addition and multiplication properties.

Problem 6. Let \(x = \begin{bmatrix} 1 \\ -1 \end{bmatrix}\). Then \(x \in \mathbb{Z}^2\), but \(\tfrac{1}{2}x \notin \mathbb{Z}^2\), so \(\mathbb{Z}^2\) is not closed under scalar multiplication.

Problem 7.

  1. Nul \(A\) is a 4-dimensional subspace of \(\mathbb{R}^{6}\).
  2. Col \(A\) is a 2-dimensional subspace of \(\mathbb{R}^{4}\).

Problem 8.

  1. 3
  2. 6

Problem 9. 0

Problem 10. 3 pivot positions. Rank \(= 3\). Nullity \(= 2\).

A basis for Row \(A\): \(\left\{ \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \end{bmatrix},\, \begin{bmatrix} 0 & 0 & 0 & 1 & 0 \end{bmatrix},\, \begin{bmatrix} 0 & 0 & 0 & 0 & 1 \end{bmatrix} \right\}\).

The first, fourth, and fifth columns of \(B\) may fail to form a basis for Col \(A\) because row operations change the column space — the pivot columns of \(B\) correspond to pivot columns of \(A\) in position, but the actual column vectors of \(B\) differ from those of \(A\).

Problem 11. \(A\) is \(7 \times 3\), so it has 3 columns. Since nullity \(= 3\), by the Rank-Nullity Theorem, rank \(= 3 - 3 = 0\). The dimension of the row space is 0 (i.e., \(A\) is the zero matrix).

Problem 12. No — the maximum number of pivot rows is \(\min(6, 3) = 3\), so the row space can have at most dimension 3. No — the null space dimension cannot exceed the number of columns (3).

Problem 13. \[[x]_\mathcal{B} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}\]

Problem 14. \[x = \begin{bmatrix} -3 \\ 6 \end{bmatrix}\]

Problem 15.

  1. True
  2. True
  3. True
  4. True
  5. False — \(\det A^T = \det A\), not \(-\det A\).
  6. True
  7. True
  8. False — \(Ax = 0\) has only the trivial solution since \(A\) is invertible.
  9. True
  10. True (Nul \(A = \{0\}\))

Problem 16. Not enough information to determine the number of rows of \(A\), but we do know there are exactly 2 linearly independent rows. For example, both \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}\) and \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}\) have 2-dimensional row spaces in \(\mathbb{R}^3\).

Problem 17. Yes — \(A^T\) is invertible (by the Invertible Matrix Theorem), so rank \(A^T = n\).

Problem 18. \(\det B = 84\)

Problem 19.

\[\text{Nul}\, A = \text{Span}\left\{ \begin{bmatrix} -2 \\ -1 \\ 1 \\ 0 \end{bmatrix} \right\}\]

\[\text{Col}\, A = \text{Span}\left\{ \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix},\, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix},\, \begin{bmatrix} 2 \\ 4 \\ 3 \end{bmatrix} \right\}\]

Problem 20.

  1. True
  2. False — \(\det 5A = 5^3 \det A = 125 \det A\).
  3. True
  4. False — a plane in \(\mathbb{R}^3\) does not necessarily contain the zero vector, so it need not be a subspace.
  5. True
  6. True
  7. True
  8. True
  9. False — a row of zeros is possible; rank \(A \leq 3\).

Problem 21.

\[x_1 = \frac{55}{-1} = -55, \qquad x_2 = \frac{39}{-1} = -39\]