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1
Row Reduction and Echelon Forms
오답
Suppose a \(3 \times 5\) coefficient matrix for a system has three pivot columns. Is the system consistent? Why or why not?
내 답안
(미작성)
정답
| 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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해설 없음
2
Row Reduction and Echelon Forms
오답
Give an example of an inconsistent underdetermined system of two equations in three unknowns.
내 답안
(미작성)
정답
Example: x1 + x2 + x3 = 1, x1 + x2 + x3 = 2
해설 없음
3
Row Reduction and Echelon Forms
오답
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.
내 답안
(미작성)
정답
Write equations and verify solution from Exercise 11
해설 없음
4
Linear Independence
오답
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.]
내 답안
(미작성)
정답
Since col1 + 3*col2 = col3, we have A*[1; 3; -1] = 0. So x = (1, 3, -1) is a nontrivial solution.
해설 없음
5
The Matrix Equation Ax = b
오답
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.
\(\begin{pmatrix} 2 \\ 6 \\ -1 \end{pmatrix} \vec{1, -1}\)
내 답안
(미작성)
정답
Undefined. The matrix has 1 column but the vector has 2 entries.
해설
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
오답
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.
내 답안
(미작성)
정답
List all possible echelon forms of a nonzero 2x2 matrix
해설 없음
7
The Matrix Equation Ax = b
오답
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.
내 답안
(미작성)
정답
True
해설
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
오답
Write a vector equation that is equivalent to the given system of equations.
\(4 x_1 + x_2 + 3 x_3 = 9\)
\(x_1 - 7 x_2 - 2 x_3 = 2\)
\(8 x_1 + 6 x_2 - 5 x_3 = 15\)
내 답안
(미작성)
정답
Write as x1 a1 + x2 a2 + x3 a3 = b vector equation
해설 없음
9
Vector Equations
오답
(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.
내 답안
(미작성)
정답
True. The line through u and the origin is {tu : t in R}, and tu = tu + 0v is in Span{u,v}.
해설 없음
10
Linear Independence
오답
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 \(S\).
내 답안
(미작성)
정답
False. If S is linearly dependent, at least one vector is a linear combination of the others, but not necessarily every vector.
해설 없음