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Chapter 1: Problem 21

Let \(B\) and \(C\) be \(n \times n\) matrices with the property that \(B \mathbf{x}=C \mathbf{x}\) for all \(\mathbf{x} \in \mathbb{R}^{n} .\) Show that \(B=C\)

Short Answer

Expert verified
To show that \(B = C\), we first write the equation \((B - C) \mathbf{x} = \mathbf{0}\). We then substitute the standard basis vector \(e_i\) for the vector \(\mathbf{x}\), which results in \((b_i - c_i) = \mathbf{0}\), where \(b_i\) and \(c_i\) are the \(i\)-th columns of matrices \(B\) and \(C\), respectively. Since this holds for all basis vectors \(e_i\), we find that \(b_i = c_i\) for all \(i\), which implies that the matrices \(B\) and \(C\) are equal. Therefore, when the property \(B \mathbf{x} = C \mathbf{x}\) holds for all \(\mathbf{x} \in \mathbb{R}^n\), we have shown that \(B = C\).

Step by step solution

01

Define the problem with the given property

Since we know that for all \(\mathbf{x} \in \mathbb{R}^{n}\), \(B\mathbf{x}=C\mathbf{x}\), we can also write the equation as: \(B \mathbf{x} - C \mathbf{x} = \mathbf{0}\) or \((B - C) \mathbf{x} = \mathbf{0}\) in which \(\mathbf{0}\) is the zero vector.
02

Analyzing the property with basis vectors

To show that \(B = C\), we will prove that their columns are equal. Let \(e_i\) be the \(i\)-th standard basis vector in \(\mathbb{R}^n\) (it has a \(1\) in the \(i\)-th position and \(0\)s in all other positions), and denote \(b_i\) and \(c_i\) as the \(i\)-th columns of matrices \(B\) and \(C\), respectively.
03

Applying the property to basis vectors

We will apply the given property to the basis vector \(e_i\) and see how it affects the column vectors of the matrices \(B\) and \(C\). Substitute \(e_i\) for the vector \(\mathbf{x}\) in the equation: \((B - C)e_i = \mathbf{0}\) Now, we know that \(Be_i = b_i\) and \(Ce_i = c_i\). So, \((b_i - c_i) = \mathbf{0}\)
04

Show that columns of B and C are equal

From the previous step, we found that for each basis vector \(e_i\): \((b_i - c_i) = \mathbf{0}\) Adding \(c_i\) to both sides of the equation, we get: \(b_i = c_i\)
05

Conclude that B and C are equal

Since we have shown that for all \(i\), the \(i\)-th columns of matrices \(B\) and \(C\) are equal, i.e., \(b_i = c_i\), we can now conclude that the matrices \(B\) and \(C\) are equal: \(B = C\) So, when the property \(B \mathbf{x} = C \mathbf{x}\) holds for all \(\mathbf{x} \in \mathbb{R}^n\), we have shown that \(B = C\).

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Most popular questions from this chapter

Let \(A\) and \(B\) be \(n \times n\) matrices and let \(C=A B\) Prove that if \(B\) is singular, then \(C\) must be singular [Hint: Use Theorem \(1.5 .2 .\)

Let \(A=\left(\begin{array}{lll}1 & 2 & 4 \\ 2 & 1 & 3 \\ 1 & 0 & 2\end{array}\right), \quad B=\left(\begin{array}{lll}1 & 2 & 4 \\ 2 & 1 & 3 \\\ 2 & 2 & 6\end{array}\right)\) \(C=\left(\begin{array}{rrr}1 & 2 & 4 \\ 0 & -1 & -3 \\ 2 & 2 & 6\end{array}\right)\) (a) Find an elementary matrix \(E\) such that \(E A=B\) (b) Find an elementary matrix \(F\) such that \(F B=C\) (c) Is \(C\) row equivalent to \(A\) ? Explain.

Let \(A\) be an \(m \times n\) matrix, \(X\) an \(n \times r\) matrix, and \(B\) an \(m \times r\) matrix. Show that \\[ A X=B \\] if and only if \\[ A \mathbf{x}_{j}=\mathbf{b}_{j}, \quad j=1, \ldots, r \\]

Let \\[ \begin{aligned} I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right], \quad E=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad O=\left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right) \\ C &=\left[\begin{array}{rr} 1 & 0 \\ -1 & 1 \end{array}\right], \quad D=\left[\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right] \end{aligned} \\] and \(B=\left[\begin{array}{ll}B_{11} & B_{12} \\ B_{21} & B_{22}\end{array}\right]=\left(\begin{array}{cc|cc}1 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 \\ \hline 3 & 1 & 1 & 1 \\ 3 & 2 & 1 & 2\end{array}\right)\) Perform each of the following block multiplications: (a) \(\left[\begin{array}{ll}O & I \\ I & O\end{array}\right]\left[\begin{array}{ll}B_{11} & B_{12} \\ B_{21} & B_{22}\end{array}\right]\) (b) \(\left[\begin{array}{ll}C & O \\ O & C\end{array}\right]\left[\begin{array}{ll}B_{11} & B_{12} \\ B_{21} & B_{22}\end{array}\right]\) (c) \(\left[\begin{array}{cc}D & O \\ O & I\end{array}\right]\left(\begin{array}{ll}B_{11} & B_{12} \\ B_{21} & B_{22}\end{array}\right)\) (d) \(\left(\begin{array}{ll}E & O \\ O & E\end{array}\right)\left(\begin{array}{ll}B_{11} & B_{12} \\ B_{21} & B_{22}\end{array}\right)\)

For each of the pairs of matrices that follow, determine whether it is possible to multiply the first matrix times the second. If it is possible, perform the multiplication. (a) \(\left[\begin{array}{rrr}3 & 5 & 1 \\ -2 & 0 & 2\end{array}\right]\left[\begin{array}{ll}2 & 1 \\ 1 & 3 \\ 4 & 1\end{array}\right]\) (b) \(\left[\begin{array}{cc}4 & -2 \\ 6 & -4 \\ 8 & -6\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3\end{array}\right]\) (c) \(\left[\begin{array}{lll}1 & 4 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 2\end{array}\right]\left[\begin{array}{ll}3 & 2 \\ 1 & 1 \\ 4 & 5\end{array}\right]\) (d) \(\left[\begin{array}{ll}4 & 6 \\ 2 & 1\end{array}\right]\left[\begin{array}{lll}3 & 1 & 5 \\ 4 & 1 & 6\end{array}\right]\) (e) \(\left[\begin{array}{lll}4 & 6 & 1 \\ 2 & 1 & 1\end{array}\right]\left[\begin{array}{lll}3 & 1 & 5 \\ 4 & 1 & 6\end{array}\right]\) (f) \(\left[\begin{array}{r}2 \\ -1 \\\ 3\end{array}\right]\left[\begin{array}{llll}3 & 2 & 4 & 5\end{array}\right]\)
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