document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165604>Question 1165604</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following functions T : R → R . Show that each is  a linear transformation and determine for each the matrix A such that T (x) = A x<BR>\n" );
document.write( "a)       x1.            x1+2x2 +3x3<BR>\n" );
document.write( "    T    x2.  =        2x2-3x1+x3<BR>\n" );
document.write( "¶         x3\r<BR>\n" );
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document.write( "b)       x1.            7x1+2x2 +x3<BR>\n" );
document.write( "   T     x2.  =        3x1-11x2+2x3<BR>\n" );
document.write( "¶        x3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853085 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=853085\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This question asks you to demonstrate that the given transformations $T: \mathbb{R}^3 \to \mathbb{R}^3$ are linear and to find the standard matrix $A$ for each, such that $T(\mathbf{x}) = A\mathbf{x}$.\r<BR>\n" );
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document.write( "## General Definition of a Linear Transformation\r<BR>\n" );
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document.write( "A transformation $T: V \to W$ is linear if, for all vectors $\mathbf{u}, \mathbf{v}$ in $V$ and all scalars $c$ in $\mathbb{R}$:<BR>\n" );
document.write( "1.  **Additivity:** $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$<BR>\n" );
document.write( "2.  **Homogeneity:** $T(c\mathbf{u}) = cT(\mathbf{u})$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## Part (a)\r<BR>\n" );
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document.write( "The transformation is defined by:<BR>\n" );
document.write( "$$T \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} x_1 + 2x_2 + 3x_3 \\ -3x_1 + 2x_2 + x_3 \\ 0 \end{pmatrix}$$<BR>\n" );
document.write( "*(Note: I assumed the third component is $0$ as the input has three components but the output only shows two distinct linear expressions, and a $3 \times 3$ matrix is expected for $T: \mathbb{R}^3 \to \mathbb{R}^3$. I'll proceed by assuming the third component is a constant like 0 or $x_1$ from the format)*\r<BR>\n" );
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document.write( "**Based on the layout, it is most likely that the output is meant to be a $3 \times 1$ column vector. Given the formatting ambiguity, I will use the coefficients as presented, and assume the third component is 0 for the linear combination interpretation.**\r<BR>\n" );
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document.write( "Let's re-write the output as a $3 \times 1$ vector where the structure dictates the coefficients:<BR>\n" );
document.write( "$$T \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} x_1 + 2x_2 + 3x_3 \\ -3x_1 + 2x_2 + x_3 \\ 0 \end{pmatrix}$$\r<BR>\n" );
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document.write( "### A. Finding the Standard Matrix $A$\r<BR>\n" );
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document.write( "The standard matrix $A$ is formed by applying $T$ to the standard basis vectors $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$:<BR>\n" );
document.write( "$$A = [T(\mathbf{e}_1) \mid T(\mathbf{e}_2) \mid T(\mathbf{e}_3)]$$\r<BR>\n" );
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document.write( "1.  $$T\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1(1) + 2(0) + 3(0) \\ -3(1) + 2(0) + 1(0) \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ -3 \\ 0 \end{pmatrix}$$<BR>\n" );
document.write( "2.  $$T\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1(0) + 2(1) + 3(0) \\ -3(0) + 2(1) + 1(0) \\ 0 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \\ 0 \end{pmatrix}$$<BR>\n" );
document.write( "3.  $$T\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1(0) + 2(0) + 3(1) \\ -3(0) + 2(0) + 1(1) \\ 0 \end{pmatrix} = \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix}$$\r<BR>\n" );
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document.write( "The standard matrix $A$ is:<BR>\n" );
document.write( "$$\mathbf{A = \begin{pmatrix} 1 & 2 & 3 \\ -3 & 2 & 1 \\ 0 & 0 & 0 \end{pmatrix}}$$\r<BR>\n" );
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document.write( "### B. Proving Linearity\r<BR>\n" );
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document.write( "Since the components of $T(\mathbf{x})$ are **linear homogeneous polynomials** (polynomials where every term has degree one) in the variables $x_1, x_2, x_3$ and there is no constant term, $T$ is automatically a linear transformation.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## Part (b)\r<BR>\n" );
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document.write( "The transformation is defined by:<BR>\n" );
document.write( "$$T \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 7x_1 + 2x_2 + x_3 \\ 3x_1 - 11x_2 + 2x_3 \end{pmatrix}$$\r<BR>\n" );
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document.write( "*(Note: In this case, $T: \mathbb{R}^3 \to \mathbb{R}^2$, as the output vector has 2 components.)*\r<BR>\n" );
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document.write( "### A. Finding the Standard Matrix $A$\r<BR>\n" );
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document.write( "The standard matrix $A$ is a $2 \times 3$ matrix formed by applying $T$ to the standard basis vectors $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$:<BR>\n" );
document.write( "$$A = [T(\mathbf{e}_1) \mid T(\mathbf{e}_2) \mid T(\mathbf{e}_3)]$$\r<BR>\n" );
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document.write( "1.  $$T\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 7(1) + 2(0) + 1(0) \\ 3(1) - 11(0) + 2(0) \end{pmatrix} = \begin{pmatrix} 7 \\ 3 \end{pmatrix}$$<BR>\n" );
document.write( "2.  $$T\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 7(0) + 2(1) + 1(0) \\ 3(0) - 11(1) + 2(0) \end{pmatrix} = \begin{pmatrix} 2 \\ -11 \end{pmatrix}$$<BR>\n" );
document.write( "3.  $$T\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 7(0) + 2(0) + 1(1) \\ 3(0) - 11(0) + 2(1) \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$$\r<BR>\n" );
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document.write( "The standard matrix $A$ is:<BR>\n" );
document.write( "$$\mathbf{A = \begin{pmatrix} 7 & 2 & 1 \\ 3 & -11 & 2 \end{pmatrix}}$$\r<BR>\n" );
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document.write( "### B. Proving Linearity\r<BR>\n" );
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document.write( "Since the components of $T(\mathbf{x})$ are again **linear homogeneous polynomials** in $x_1, x_2, x_3$ (no terms with degree other than one and no constant terms), $T$ is a linear transformation.\r<BR>\n" );
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document.write( "**Formal Proof for Additivity (Homogeneity is similar):**\r<BR>\n" );
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document.write( "Let $\mathbf{u} = \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$.\r<BR>\n" );
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document.write( "$$T(\mathbf{u} + \mathbf{v}) = T \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix}$$\r<BR>\n" );
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document.write( "$$T(\mathbf{u} + \mathbf{v}) = \begin{pmatrix} 7(u_1 + v_1) + 2(u_2 + v_2) + (u_3 + v_3) \\ 3(u_1 + v_1) - 11(u_2 + v_2) + 2(u_3 + v_3) \end{pmatrix}$$\r<BR>\n" );
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document.write( "Rearranging the terms based on $\mathbf{u}$ and $\mathbf{v}$:\r<BR>\n" );
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document.write( "$$T(\mathbf{u} + \mathbf{v}) = \begin{pmatrix} (7u_1 + 2u_2 + u_3) + (7v_1 + 2v_2 + v_3) \\ (3u_1 - 11u_2 + 2u_3) + (3v_1 - 11v_2 + 2v_3) \end{pmatrix}$$\r<BR>\n" );
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document.write( "$$T(\mathbf{u} + \mathbf{v}) = \begin{pmatrix} 7u_1 + 2u_2 + u_3 \\ 3u_1 - 11u_2 + 2u_3 \end{pmatrix} + \begin{pmatrix} 7v_1 + 2v_2 + v_3 \\ 3v_1 - 11v_2 + 2v_3 \end{pmatrix} = T(\mathbf{u}) + T(\mathbf{v})$$\r<BR>\n" );
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document.write( "Thus, $T$ is linear. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );