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1
Row Reduction and Echelon Forms
Wrong
Suppose a \(3 \times 5\) coefficient matrix for a system has three pivot columns. Is the system consistent? Why or why not?
My Answer
(No answer)
Correct Answer
| Yes, consistent. Three pivots in a 3x5 coefficient matrix means every row has a pivot, so no row of the form [0 0 0 0 0 | b] with b nonzero can occur. |
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No explanation
2
Row Reduction and Echelon Forms
Wrong
Give an example of an inconsistent underdetermined system of two equations in three unknowns.
My Answer
(No answer)
Correct Answer
Example: x1 + x2 + x3 = 1, x1 + x2 + x3 = 2
No explanation
3
Row Reduction and Echelon Forms
Wrong
Write down the equations corresponding to the augmented matrix in Exercise 11 and verify that your answer to Exercise 11 is correct by substituting the solutions you obtained back into the original equations.
My Answer
(No answer)
Correct Answer
Write equations and verify solution from Exercise 11
No explanation
4
Linear Independence
Wrong
Explain why the columns of the matrix \(A\) are linearly dependent, and find a nontrivial solution of \(A \mathbf{x} = \mathbf{0}\) without row operations.
\(A = \begin{pmatrix} 5 & 1 & 8 \\ -9 & 5 & 6 \\ 6 & -5 & -9 \end{pmatrix}\)
[Note: The first column plus 3 times the second column equals the third column.]
My Answer
(No answer)
Correct Answer
Since col1 + 3*col2 = col3, we have A*[1; 3; -1] = 0. So x = (1, 3, -1) is a nontrivial solution.
No explanation
5
The Matrix Equation Ax = b
Wrong
Compute the products using (a) the definition, as in Example 1, and using (b) the row–vector rule for computing \(A \mathbf{x}\). If a product is undefined, explain why.
\(\begin{pmatrix} 2 \\ 6 \\ -1 \end{pmatrix} \vec{1, -1}\)
My Answer
(No answer)
Correct Answer
Undefined. The matrix has 1 column but the vector has 2 entries.
Explanation
The matrix is \(3 \times 1\) and the vector is in \(RR^2\). The number of columns of the matrix (1) does not match the number of entries in the vector (2), so the product is undefined.
6
Row Reduction and Echelon Forms
Wrong
Describe the possible echelon forms of a nonzero \(2 \times 2\) matrix. Use the symbols \(\mathbf{square.filled}\), \(*\), and \(0\), as in the first part of Example 1.
My Answer
(No answer)
Correct Answer
List all possible echelon forms of a nonzero 2x2 matrix
No explanation
7
The Matrix Equation Ax = b
Wrong
Mark each statement True or False. Justify each answer.
If \(A\) is an \(m \times n\) matrix and if the equation \(A \mathbf{x} = \mathbf{b}\) is inconsistent for some \(\mathbf{b}\) in \(RR^m\), then \(A\) cannot have a pivot position in every row.
My Answer
(No answer)
Correct Answer
True
Explanation
By the contrapositive of Theorem 4: if \(A\) had a pivot in every row, then \(A \mathbf{x} = \mathbf{b}\) would be consistent for all \(\mathbf{b}\).
8
Vector Equations
Wrong
Write a vector equation that is equivalent to the given system of equations.
\(4 x_1 + x_2 + 3 x_3 = 9\)
\(x_1 - 7 x_2 - 2 x_3 = 2\)
\(8 x_1 + 6 x_2 - 5 x_3 = 15\)
My Answer
(No answer)
Correct Answer
Write as x1 a1 + x2 a2 + x3 a3 = b vector equation
No explanation
9
Vector Equations
Wrong
(T/F) When \(\mathbf{u}\) and \(\mathbf{v}\) are nonzero vectors, \(\text{Span} {\mathbf{u}, \mathbf{v}}\) contains the line through \(\mathbf{u}\) and the origin.
My Answer
(No answer)
Correct Answer
True. The line through u and the origin is {tu : t in R}, and tu = tu + 0v is in Span{u,v}.
No explanation
10
Linear Independence
Wrong
Mark each statement True or False. Justify each answer.
(b) If \(S\) is a linearly dependent set, then each vector is a linear combination of the other vectors in \(S\).
My Answer
(No answer)
Correct Answer
False. If S is linearly dependent, at least one vector is a linear combination of the others, but not necessarily every vector.
No explanation