document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165864>Question 1165864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Kuttler, pg. 279, Ex. 5.2.8) Consider the following functions T : R3→ R2 . Show that each is a linear transformation and determine for each the matrix A such that <BR>\n" );
document.write( "T (x) = A x.<BR>\n" );
document.write( "T . (matrix3x1(x1, x2, x3) )= Matrix2x1(x1+2x2+3x3, 2x2−3x1+x3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852880 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2103)</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=852880\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem asks you to confirm that a given function $T: \mathbb{R}^3 \to \mathbb{R}^2$ is a **linear transformation** and then find its **standard matrix** $A$.\r<BR>\n" );
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document.write( "The given transformation is:<BR>\n" );
document.write( "$$T \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} x_1 + 2x_2 + 3x_3 \\ -3x_1 + 2x_2 + x_3 \end{pmatrix}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 1. Showing $T$ is a Linear Transformation\r<BR>\n" );
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document.write( "A function $T$ is a linear transformation if it satisfies two properties for any vectors $\mathbf{u}$ and $\mathbf{v}$ in the domain ($\mathbb{R}^3$) and any scalar $c$ in $\mathbb{R}$:\r<BR>\n" );
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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( "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( "### 1.1. Additivity\r<BR>\n" );
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document.write( "$$\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix}$$\r<BR>\n" );
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document.write( "$$\begin{aligned} T(\mathbf{u} + \mathbf{v}) &= T \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix} \\ &= \begin{pmatrix} (u_1+v_1) + 2(u_2+v_2) + 3(u_3+v_3) \\ -3(u_1+v_1) + 2(u_2+v_2) + (u_3+v_3) \end{pmatrix} \\ &= \begin{pmatrix} (u_1 + 2u_2 + 3u_3) + (v_1 + 2v_2 + 3v_3) \\ (-3u_1 + 2u_2 + u_3) + (-3v_1 + 2v_2 + v_3) \end{pmatrix} \\ &= \begin{pmatrix} u_1 + 2u_2 + 3u_3 \\ -3u_1 + 2u_2 + u_3 \end{pmatrix} + \begin{pmatrix} v_1 + 2v_2 + 3v_3 \\ -3v_1 + 2v_2 + v_3 \end{pmatrix} \\ &= T(\mathbf{u}) + T(\mathbf{v}) \end{aligned}$$<BR>\n" );
document.write( "**Additivity is satisfied.**\r<BR>\n" );
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document.write( "### 1.2. Homogeneity\r<BR>\n" );
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document.write( "$$c\mathbf{u} = \begin{pmatrix} cu_1 \\ cu_2 \\ cu_3 \end{pmatrix}$$\r<BR>\n" );
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document.write( "$$\begin{aligned} T(c\mathbf{u}) &= T \begin{pmatrix} cu_1 \\ cu_2 \\ cu_3 \end{pmatrix} \\ &= \begin{pmatrix} (cu_1) + 2(cu_2) + 3(cu_3) \\ -3(cu_1) + 2(cu_2) + (cu_3) \end{pmatrix} \\ &= \begin{pmatrix} c(u_1 + 2u_2 + 3u_3) \\ c(-3u_1 + 2u_2 + u_3) \end{pmatrix} \\ &= c \begin{pmatrix} u_1 + 2u_2 + 3u_3 \\ -3u_1 + 2u_2 + u_3 \end{pmatrix} \\ &= cT(\mathbf{u}) \end{aligned}$$<BR>\n" );
document.write( "**Homogeneity is satisfied.**\r<BR>\n" );
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document.write( "Since both properties hold, $T$ is a **linear transformation**. (In general, any transformation defined by a system of linear equations with no constant terms is a linear transformation.)\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 2. Determining the Standard Matrix $A$\r<BR>\n" );
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document.write( "The standard matrix $A$ of a linear transformation $T$ is the matrix whose columns are the images of the **standard basis vectors** of the domain ($\mathbb{R}^3$):<BR>\n" );
document.write( "$$A = \begin{bmatrix} T(\mathbf{e}_1) & T(\mathbf{e}_2) & T(\mathbf{e}_3) \end{bmatrix}$$<BR>\n" );
document.write( "where $\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, $\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$, and $\mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$.\r<BR>\n" );
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document.write( "1.  **Image of $\mathbf{e}_1$:**<BR>\n" );
document.write( "    $$T \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} (1) + 2(0) + 3(0) \\ -3(1) + 2(0) + (0) \end{pmatrix} = \begin{pmatrix} 1 \\ -3 \end{pmatrix}$$\r<BR>\n" );
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document.write( "2.  **Image of $\mathbf{e}_2$:**<BR>\n" );
document.write( "    $$T \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} (0) + 2(1) + 3(0) \\ -3(0) + 2(1) + (0) \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \end{pmatrix}$$\r<BR>\n" );
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document.write( "3.  **Image of $\mathbf{e}_3$:**<BR>\n" );
document.write( "    $$T \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} (0) + 2(0) + 3(1) \\ -3(0) + 2(0) + (1) \end{pmatrix} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$$\r<BR>\n" );
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document.write( "### The Matrix $A$\r<BR>\n" );
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document.write( "Assembling the column vectors:<BR>\n" );
document.write( "$$A = \begin{pmatrix} 1 & 2 & 3 \\ -3 & 2 & 1 \end{pmatrix}$$\r<BR>\n" );
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document.write( "We can verify that $T(\mathbf{x}) = A\mathbf{x}$:<BR>\n" );
document.write( "$$A\mathbf{x} = \begin{pmatrix} 1 & 2 & 3 \\ -3 & 2 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 1x_1 + 2x_2 + 3x_3 \\ -3x_1 + 2x_2 + 1x_3 \end{pmatrix} = T(\mathbf{x})$$\r<BR>\n" );
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document.write( "The standard matrix is:<BR>\n" );
document.write( "$$A = \begin{pmatrix} 1 & 2 & 3 \\ -3 & 2 & 1 \end{pmatrix}$$ </SPAN>\n" );
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