document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165858>Question 1165858</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let matrix3x2 A = 1, -1 , 2, 0, 3, -4. Determine whether multiplication by A is a one-to-one transformation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852881 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852881\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To determine whether multiplication by a matrix $A$ is a **one-to-one transformation**, we need to analyze the properties of the linear transformation $T(\mathbf{x}) = A\mathbf{x}$.\r<BR>\n" );
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document.write( "A linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ is **one-to-one (injective)** if and only if **$T(\mathbf{x}) = \mathbf{0}$ has only the trivial solution** ($\mathbf{x} = \mathbf{0}$). Equivalently, this means the **null space (kernel)** of $A$ contains only the zero vector, or that the matrix $A$ has **linearly independent columns**.\r<BR>\n" );
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document.write( "## 1. Analyze the Matrix and Transformation\r<BR>\n" );
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document.write( "The given matrix $A$ is a $3 \times 2$ matrix (3 rows, 2 columns):<BR>\n" );
document.write( "$$A = \begin{pmatrix} 1 & -1 \\ 2 & 0 \\ 3 & -4 \end{pmatrix}$$\r<BR>\n" );
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document.write( "The transformation $T$ maps vectors from $\mathbb{R}^2$ to $\mathbb{R}^3$, i.e., $T: \mathbb{R}^2 \to \mathbb{R}^3$.<BR>\n" );
document.write( "$$T \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ 2 & 0 \\ 3 & -4 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$\r<BR>\n" );
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document.write( "## 2. Check for One-to-One Property\r<BR>\n" );
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document.write( "We check the condition by solving the homogeneous equation $A\mathbf{x} = \mathbf{0}$:\r<BR>\n" );
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document.write( "$$\begin{pmatrix} 1 & -1 \\ 2 & 0 \\ 3 & -4 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$\r<BR>\n" );
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document.write( "This corresponds to the system of equations:<BR>\n" );
document.write( "1. $x_1 - x_2 = 0$<BR>\n" );
document.write( "2. $2x_1 + 0x_2 = 0$<BR>\n" );
document.write( "3. $3x_1 - 4x_2 = 0$\r<BR>\n" );
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document.write( "From equation (2):<BR>\n" );
document.write( "$$2x_1 = 0 \implies x_1 = 0$$\r<BR>\n" );
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document.write( "Substitute $x_1 = 0$ into equation (1):<BR>\n" );
document.write( "$$0 - x_2 = 0 \implies x_2 = 0$$\r<BR>\n" );
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document.write( "Substitute $x_1 = 0$ and $x_2 = 0$ into equation (3) to verify consistency:<BR>\n" );
document.write( "$$3(0) - 4(0) = 0 \implies 0 = 0$$\r<BR>\n" );
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document.write( "Since the only solution to $A\mathbf{x} = \mathbf{0}$ is the **trivial solution** ($\mathbf{x} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$), the columns of $A$ are **linearly independent**.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## Conclusion\r<BR>\n" );
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document.write( "Yes, multiplication by matrix $A$ **is a one-to-one transformation**. </SPAN>\n" );
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