document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166343>Question 1166343</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 6 Let V = C[a, b] be the vector space of continuous functions on interval <BR>\n" );
document.write( "[a, b]. Let W be the subset of functions in V such that:\r<BR>\n" );
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document.write( "(integralsign) from(a,b) f(x) dx = 0.<BR>\n" );
document.write( "Is W a subspace of V ? You must justify your answer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852764 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=852764\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Yes, **W** is a **subspace** of $\mathbf{V}$, the vector space of continuous functions on the interval $[a, b]$.\r<BR>\n" );
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document.write( "The vector space $V$ is $C[a, b]$, the set of all continuous real-valued functions on the interval $[a, b]$. The subset $W$ is defined as:<BR>\n" );
document.write( "$$W = \left\{ f(x) \in V \ \bigg| \ \int_a^b f(x) \ dx = 0 \right\}$$\r<BR>\n" );
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document.write( "To prove that $W$ is a subspace of $V$, we must verify the three conditions:\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 1. The Zero Vector Condition (Contains the Zero Function)\r<BR>\n" );
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document.write( "The zero vector in $V$ is the **zero function**, $\mathbf{z}(x) = 0$, for all $x \in [a, b]$.\r<BR>\n" );
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document.write( "We must check if the zero function satisfies the condition for $W$:<BR>\n" );
document.write( "$$\int_a^b \mathbf{z}(x) \ dx = \int_a^b 0 \ dx = 0$$<BR>\n" );
document.write( "Since the integral of the zero function is $0$, the zero function is in $W$.<BR>\n" );
document.write( "Thus, $W$ is non-empty.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 2. Closure under Vector Addition ➕\r<BR>\n" );
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document.write( "Let $f(x)$ and $g(x)$ be two arbitrary functions in $W$. This means:<BR>\n" );
document.write( "* $\int_a^b f(x) \ dx = 0$<BR>\n" );
document.write( "* $\int_a^b g(x) \ dx = 0$\r<BR>\n" );
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document.write( "We need to check if their sum, $(f+g)(x) = f(x) + g(x)$, is also in $W$. We check the integral of the sum:<BR>\n" );
document.write( "$$\int_a^b (f(x) + g(x)) \ dx = \int_a^b f(x) \ dx + \int_a^b g(x) \ dx \quad \text{(Property of Integrals)}$$<BR>\n" );
document.write( "Substitute the conditions for $f$ and $g$:<BR>\n" );
document.write( "$$\int_a^b (f(x) + g(x)) \ dx = 0 + 0 = 0$$<BR>\n" );
document.write( "Since the integral of the sum is $0$, the function $(f+g)(x)$ is in $W$.<BR>\n" );
document.write( "Thus, $W$ is closed under function addition.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 3. Closure under Scalar Multiplication ✖️\r<BR>\n" );
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document.write( "Let $f(x)$ be an arbitrary function in $W$ (so $\int_a^b f(x) \ dx = 0$), and let $c$ be an arbitrary scalar (a real number).\r<BR>\n" );
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document.write( "We need to check if the scalar multiple, $(cf)(x) = c \cdot f(x)$, is also in $W$. We check the integral of the scalar multiple:<BR>\n" );
document.write( "$$\int_a^b c \cdot f(x) \ dx = c \cdot \int_a^b f(x) \ dx \quad \text{(Property of Integrals)}$$<BR>\n" );
document.write( "Substitute the condition for $f$:<BR>\n" );
document.write( "$$\int_a^b c \cdot f(x) \ dx = c \cdot (0) = 0$$<BR>\n" );
document.write( "Since the integral of the scalar multiple is $0$, the function $(cf)(x)$ is in $W$.<BR>\n" );
document.write( "Thus, $W$ is closed under scalar multiplication.\r<BR>\n" );
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document.write( "***\r<BR>\n" );
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document.write( "Since $W$ satisfies all three conditions, it is a **subspace** of the vector space $V$.\r<BR>\n" );
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document.write( "Would you like to analyze a different set of continuous functions to see if it forms a subspace? </SPAN>\n" );
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