4
Answer the following problem:
Let \(A\) be a \(4 \times 3\) matrix. Denote by \(C_i\) the \(i\)-th column of \(A\), and let \(b=C_1-2C_3.\)
(a) Prove that the system \(A x = b\) has a solution.
Suppose \(A=\begin{pmatrix}a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ a_{41} & a_{42} & a_{43}\end{pmatrix}\), then \(Ax=b\Rightarrow\begin{pmatrix}a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ a_{41} & a_{42} & a_{43}\end{pmatrix}\begin{pmatrix}x_1\\ x_2\\ x_3\end{pmatrix}=\begin{pmatrix}a_{11}-2a_{13}\\ a_{21}-2a_{23}\\ a_{31}-2a_{33}\\ a_{41}-2a_{43}\end{pmatrix}\)
Then we have \(\begin{cases}a_{11}x_1+a_{12}x_2+a_{13}x_3=a_{11}-2a_{13}\\ a_{21}x_1+a_{22}x_2+a_{23}x_3=a_{21}-2a_{23}\\ a_{31}x_1+a_{32}x_2+a_{33}x_3=a_{31}-2a_{33}\\ a_{41}x_1+a_{42}x_2+a_{43}x_3=a_{41}-2a_{43}\end{cases}\)
Obviously, there exists a solution which is \(x_0=\begin{pmatrix}1\\ 0\\ -2\end{pmatrix}\)
Thus the system \(A x = b\) has a solution.
(b) Assume that \(\text{rank}(A) = 2\), and that \(x_0 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\) is a solution to \(A x = b\). Prove that the system has infinitely many solutions, find the degree of freedom, and write the general solution.
Since \(\begin{cases}a_{11}x_1+a_{12}x_2+a_{13}x_3=a_{11}-2a_{13}\\ a_{21}x_1+a_{22}x_2+a_{23}x_3=a_{21}-2a_{23}\\ a_{31}x_1+a_{32}x_2+a_{33}x_3=a_{31}-2a_{33}\\ a_{41}x_1+a_{42}x_2+a_{43}x_3=a_{41}-2a_{43}\end{cases}\) and \(x_0 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\), then \(\begin{cases}a_{11}+a_{12}+a_{13}=a_{11}-2a_{13}\\ a_{21}+a_{22}+a_{23}=a_{21}-2a_{23}\\ a_{31}+a_{32}+a_{33}=a_{31}-2a_{33}\\ a_{41}+a_{42}+a_{43}=a_{41}-2a_{43}\end{cases}\)
Then \(\begin{cases}a_{12}=-3a_{13}\\ a_{22}=-3a_{23}\\ a_{32}=-3a_{33}\\ a_{42}=-3a_{43}\end{cases}\), Let's consider \(A\) be the row-reduced echelon form, thus the first column must have a pivot which is \(a_{11}=1\)
Then \(a_{i2}=-3a_{i3}\), since \(a_{i2}\) is in the front of \(a_{i3}\), then \(\exists a_{i2}=1\)
Since \(rank(A)=2\), then \(\exists!a_{i2}=1,a_{i3}=-\frac13\)
Since \(A\) is the row-reduced echelon form, then \(a_{22}=1,a_{23}=-\frac13\) and \(a_{12},a_{32},a_{42}=0\Rightarrow a_{13},a_{33},a_{43}=0\)
We have \(A=\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & -\frac13\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix}\), then \(\begin{cases}x_1=1\\ x_2-\frac13x_3=\frac23\end{cases}\)
Then \(x_3\) can be any values, thus the system has infinitely many solutions
The degree of freedom is 1
The general solution is \(\begin{cases}x_1=1\\ x_2=\frac23+\frac13x_3\\ x_3 \end{cases}\)
Answer the following questions:
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Rate the difficulty of the given problem. \((1 - \text{very easy}, 9 - \text{very hard})\) 1 2 3 4 5 6 7 8 9
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How confident are you that you answered correctly? \((1 - \text{not confident at all}, 9 - \text{very confident})\) 1 2 3 4 5 6 7 8 9
If you think you were right, what helped you answer correctly?
Maybe the idea that lecturer taught me, i have a deep understanding of row-reduced echelon form, thus i can use it and simplify the matrix easily, and i also consider the idea of question and its meaning.
If you think you were wrong, what could have helped you answer correctly?
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Do you agree with the following statements? \((1 - \text{don't agree at all}, 9 - \text{strongly agree})\)
(a) I understand the problem's instructions and what it means to write a formal solution. 1 2 3 4 5 6 7 8 9
(b) I intuitively understand what is the solution to the problem. 1 2 3 4 5 6 7 8 9
(c) My teachers expect me to strictly follow the solution path demonstrated in class. 1 2 3 4 5 6 7 8 9
(d) My teachers encourage me to develop independent thinking. 1 2 3 4 5 6 7 8 9
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Give an example of a tip you have learned in the lectures or tutorials in order to succeed in mathematics.
If i want to examine whether i really have a good command of a mathematics question or concept. I should retell it to myself and try to teach others. If i succeed, I really understand it. I think this method is really helpful