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Linear Algebra for Team-Based Inquiry Learning 2025 Edition

Steven Clontz, Drew Lewis

Section 3.5 Vector Spaces (AT5)

Subsection 3.5.1 Warm Up

Activity 3.5.1.

(a)
How would you describe a sandwich to someone who has never seen a sandwich before?
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(b)
How would you describe to someone what a vector is?
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Subsection 3.5.2 Class Activities

Observation 3.5.2.

Consider the following applications of properties of the real numbers \(\mathbb R\text{:}\)
  1. \(1+(2+3)=(1+2)+3\text{.}\)
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  2. \(7+4=4+7\text{.}\)
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  3. There exists some \(\unknown\) where \(5+\unknown=5\text{.}\)
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  4. There exists some \(\unknown\) where \(9+\unknown=0\text{.}\)
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  5. \(\frac{1}{2}(1+7)\) is the only number that is equally distant from \(1\) and \(7\text{.}\)
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Activity 3.5.3.

Which of the following properties of \(\IR^2\) Euclidean vectors is NOT true?
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  1. \(\left[\begin{array}{c} x_1\\x_2\end{array}\right] +\left(\left[\begin{array}{c} y_1\\y_2\end{array}\right] +\left[\begin{array}{c} z_1\\z_2\end{array}\right]\right)= \left(\left[\begin{array}{c} x_1\\x_2\end{array}\right] +\left[\begin{array}{c} y_1\\y_2\end{array}\right]\right) +\left[\begin{array}{c} z_1\\z_2\end{array}\right]\text{.}\)
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  2. \(\left[\begin{array}{c}x_1\\x_2\end{array}\right] + \left[\begin{array}{c}y_1\\y_2\end{array}\right] = \left[\begin{array}{c}y_1\\y_2\end{array}\right] + \left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)
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  3. There exists some \(\left[\begin{array}{c}\unknown\\\unknown\end{array}\right]\) where \(\left[\begin{array}{c}x_1\\x_2\end{array}\right] +\left[\begin{array}{c}\unknown\\\unknown\end{array}\right] =\left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)
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  4. There exists some \(\left[\begin{array}{c}\unknown\\\unknown\end{array}\right]\) where \(\left[\begin{array}{c}x_1\\x_2\end{array}\right]+ \left[\begin{array}{c}\unknown\\\unknown\end{array}\right]= \left[\begin{array}{c}0\\0\end{array}\right]\text{.}\)
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  5. \(\displaystyle\frac{1}{2}\left(\left[\begin{array}{c}x_1\\x_2\end{array}\right] + \left[\begin{array}{c}y_1\\y_2\end{array}\right] \right)\) is the only vector whose endpoint is equally distant from the endpoints of \(\left[\begin{array}{c}x_1\\x_2\end{array}\right]\) and \(\left[\begin{array}{c}y_1\\y_2\end{array}\right]\text{.}\)
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Observation 3.5.4.

Consider the following applications of properties of the real numbers \(\mathbb R\text{:}\)
  1. \(3(2(7))=(3\cdot 2)(7)\text{.}\)
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  2. \(1(19)=19\text{.}\)
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  3. There exists some \(\unknown\) such that \(\unknown \cdot 4= 9\text{.}\)
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  4. \(3\cdot (2+8)=3\cdot 2+3\cdot 8\text{.}\)
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  5. \((2+7)\cdot 4=2\cdot 4+7\cdot 4\text{.}\)
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Activity 3.5.5.

Which of the following properties of \(\IR^2\) Euclidean vectors is NOT true?
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  1. \(a\left(b\left[\begin{array}{c}x_1\\x_2\end{array}\right]\right)= ab\left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)
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  2. \(1\left[\begin{array}{c}x_1\\x_2\end{array}\right]= \left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)
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  3. There exists some \(\unknown\) such that \(\unknown\left[\begin{array}{c}x_1\\x_2\end{array}\right]= \left[\begin{array}{c}y_1\\y_2\end{array}\right]\text{.}\)
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  4. \(a(\vec u+\vec v)=a\vec u+a\vec v\text{.}\)
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  5. \((a+b)\vec v=a\vec v+b\vec v\text{.}\)
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Definition 3.5.7.

A vector space \(V\) is any set of mathematical objects, called vectors, and a set of numbers, called scalars, with associated addition \(\oplus\) and scalar multiplication \(\odot\) operations that satisfy the following properties. Let \(\vec u,\vec v,\vec w\) be vectors belonging to \(V\text{,}\) and let \(a,b\) be scalars.
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  1. Vector addition is associative: \(\vec u\oplus (\vec v\oplus \vec w)= (\vec u\oplus \vec v)\oplus \vec w\text{.}\)
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  2. Vector addition is commutative: \(\vec u\oplus \vec v= \vec v\oplus \vec u\text{.}\)
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  3. An additive identity exists: There exists some \(\vec z\) where \(\vec v\oplus \vec z=\vec v\text{.}\)
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  4. Additive inverses exist: There exists some \(-\vec v\) where \(\vec v\oplus (-\vec v)=\vec z\text{.}\)
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  5. Scalar multiplication is associative: \(a\odot(b\odot\vec v)=(ab)\odot\vec v\text{.}\)
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  6. 1 is a multiplicative identity: \(1\odot\vec v=\vec v\text{.}\)
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  7. Scalar multiplication distributes over vector addition: \(a\odot(\vec u\oplus \vec v)=(a\odot\vec u)\oplus(a\odot\vec v)\text{.}\)
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  8. Scalar multiplication distributes over scalar addition: \((a+ b)\odot\vec v=(a\odot\vec v)\oplus(b\odot \vec v)\text{.}\)
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Remark 3.5.8.

Consider the set \(\IC\) of complex numbers with the usual definition for addition: \((a+b\mathbf i)\oplus(c+d\mathbf i)=(a+c)+(b+d)\mathbf i\text{.}\)
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Let \(\vec u=a+b\mathbf{i}\text{,}\) \(\vec v=c+d\mathbf{i}\text{,}\) and \(\vec w=e+f\mathbf{i}\text{.}\) Then
\begin{align*} \vec u\oplus(\vec v \oplus \vec w) &= (a+b\mathbf{i})\oplus((c+d\mathbf{i})\oplus(e+f\mathbf{i}))\\ &= (a+b\mathbf{i})\oplus((c+e)+(d+f)\mathbf{i})\\ &=(a+c+e)+(b+d+f)\mathbf{i} \end{align*}
\begin{align*} (\vec u\oplus\vec v) \oplus \vec w &= ((a+b\mathbf{i})\oplus(c+d\mathbf{i}))\oplus(e+f\mathbf{i})\\ &=((a+c)+(b+d)\mathbf{i})\oplus(e+f\mathbf{i})\\ &=(a+c+e)+(b+d+f)\mathbf{i} \end{align*}
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This proves that complex addition is associative: \(\vec u\oplus(\vec v \oplus \vec w) = (\vec u\oplus\vec v) \oplus \vec w\text{.}\) The seven other vector space properties may also be verified, so \(\IC\) is an example of a vector space.
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Remark 3.5.9.

The following sets are just a few examples of vector spaces, with the usual/natural operations for addition and scalar multiplication.
  • \(\IR^n\text{:}\) Euclidean vectors with \(n\) components.
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  • \(\IC\text{:}\) Complex numbers.
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  • \(M_{m,n}\text{:}\) Matrices of real numbers with \(m\) rows and \(n\) columns.
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  • \(\P_n\text{:}\) Polynomials of degree \(n\) or less.
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  • \(\P\text{:}\) Polynomials of any degree.
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  • \(C(\IR)\text{:}\) Real-valued continuous functions.
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Activity 3.5.10.

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\text{.}\)
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Which of the following vectors is not in \(V\text{?}\)
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  1. \(\displaystyle (0, 0)\)
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  2. \(\displaystyle (1, 2)\)
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  3. \(\displaystyle (2, 4)\)
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  4. \(\displaystyle (3, 8)\)
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Activity 3.5.11.

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with the operation \(\oplus\) defined by
\begin{equation*} (x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1y_2) \text{.} \end{equation*}
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Let \(\vec u, \vec v\) be in \(V\) with \(\vec u=(1, 2)\) and \(\vec v=(2, 4)\text{.}\) Using the operations defined for \(V\text{,}\) which of the following is \(\vec u\oplus\vec v\text{?}\)
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  1. \(\displaystyle (2, 6)\)
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  2. \(\displaystyle (2, 8)\)
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  3. \(\displaystyle (3, 6)\)
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  4. \(\displaystyle (3, 8)\)
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Activity 3.5.12.

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by
\begin{equation*} (x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1y_2) \hspace{3em} c\odot (x,y)=(cx,y^c)\text{.} \end{equation*}
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Let \(a=2, b=-3\) be scalars and \(\vec u=(1,2) \in V\text{.}\)
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(a)
Verify that
\begin{equation*} (a+b)\odot \vec u=\left(-1,\frac{1}{2}\right)\text{.} \end{equation*}
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(b)
Compute the value of
\begin{equation*} \left(a\odot \vec u\right)\oplus \left(b\odot \vec u\right)\text{.} \end{equation*}
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Activity 3.5.13.

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by
\begin{equation*} (x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1y_2) \hspace{3em} c\odot (x,y)=(cx,y^c)\text{.} \end{equation*}
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Let \(a, b\) be unspecified scalars in \(\mathbb R\) and \(\vec u = (x,y)\) be an unspecified vector in \(V\text{.}\)
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(a)
Show that both sides of the equation
\begin{equation*} (a+b)\odot (x,y)= \left(a\odot (x,y)\right)\oplus \left(b\odot (x,y)\right) \end{equation*}
simplify to the expression \((ax+bx,y^ay^b)\text{.}\)
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(b)
Show that \(V\) contains an additive identity element \(\vec{z}=(\unknown,\unknown)\) satisfying
\begin{equation*} (x,y)\oplus(\unknown,\unknown)=(x,y) \end{equation*}
for all \((x,y)\in V\text{.}\)
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That is, pick appropriate values for \(\vec{z}=(\unknown,\unknown)\) and then simplify \((x,y)\oplus(\unknown,\unknown)\) into just \((x,y)\text{.}\)
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Remark 3.5.14.

It turns out \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by
\begin{equation*} (x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1y_2) \hspace{3em} c\odot (x,y)=(cx,y^c) \end{equation*}
satisfies all eight properties from Definition 3.5.7.
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Thus, \(V\) is a vector space.
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Activity 3.5.15.

Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by
\begin{equation*} (x_1,y_1)\oplus (x_2,y_2)=(x_1+y_1+x_2+y_2,x_1^2+x_2^2) \end{equation*}
\begin{equation*} c\odot (x,y)=(x^c,y+c-1)\text{.} \end{equation*}
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(a)
Show that \(1\) is the scalar multiplication identity element by simplifying \(1\odot(x,y)\) to \((x,y)\text{.}\)
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(b)
Show that \(V\) does not have an additive identity element \(\vec z=(z,w)\) by showing that \((0,-1)\oplus(z,w)\not=(0,-1)\) no matter what the values of \(z,w\) are.
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Activity 3.5.16.

Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by
\begin{equation*} (x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1+3y_2) \hspace{3em} c\odot (x,y)=(cx,cy) . \end{equation*}
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(a)
Show that scalar multiplication distributes over vector addition, i.e.
\begin{equation*} c \odot \left( (x_1,y_1) \oplus (x_2,y_2) \right) = c\odot (x_1,y_1) \oplus c\odot (x_2,y_2) \end{equation*}
for all \(c\in \IR,\, (x_1,y_1),(x_2,y_2) \in V\text{.}\)
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(b)
Show that vector addition is not associative, i.e.
\begin{equation*} (x_1,y_1) \oplus \left((x_2,y_2) \oplus (x_3,y_3)\right) \neq \left((x_1,y_1)\oplus (x_2,y_2)\right) \oplus (x_3,y_3) \end{equation*}
for some vectors \((x_1,y_1), (x_2,y_2), (x_3,y_3) \in V\text{.}\)
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Subsection 3.5.3 Cooldown

Activity 3.5.17.

(a)
What are some objects that are important to you personally, academically, or otherwise that appear vector-like to you? What makes them feel vector-like? Which axiom for vector spaces does not hold for these objects, if any?
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(b)
Our vector space axioms have eight properties. While these eight properties are enough to capture vectors, the objects that we study in the real world often have additional structures not captured by these axioms. What are some structures that you have encountered in other classes, or in previous experiences, that are not captured by these eight axioms?
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Subsection 3.5.4 Videos

Figure 37. Video: Verifying that a vector space property holds
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Figure 38. Video: Showing something is not a vector space
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Subsection 3.5.5 Exercises

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Subsection 3.5.6 Mathematical Writing Explorations

Exploration 3.5.18.

  • Show that \(\mathbb{R}^+\text{,}\) the set of positive real numbers, is a vector space, but where \(x\oplus y\) really means the product (so \(2 \oplus 3 = 6\)), and where scalar multiplication \(\alpha\odot x\) really means \(x^\alpha\text{.}\) Yes, you really do need to check all of the properties, but this is the only time I’ll make you do so. Remember, examples aren’t proofs, so you should start with arbitrary elements of \(\mathbb R^+\) for your vectors. Make sure you’re careful about telling the reader what \(\alpha\) means.
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  • Prove that the additive identity \(\vec{z}\) in an arbitrary vector space is unique.
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  • Prove that additive inverses are unique. Assume you have a vector space \(V\) and some \(\vec{v} \in V\text{.}\) Further, assume \(\vec{w_1},\vec{w_2} \in V\) with \(\vec{v} \oplus \vec{w_1} = \vec{v} \oplus \vec{w_2} = \vec{z}\text{.}\) Prove that \(\vec{w_1} = \vec{w_2}\text{.}\)
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Exploration 3.5.19.

Consider the vector space of polynomials, \(\P_n\text{.}\) Suppose further that \(n= ab\text{,}\) where \(a \mbox{ and } b\) are each positive integers. Conjecture a relationship between \(M_{a,b}\) and \(\P_n\text{.}\) We will investigate this further in section Section 3.6
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Subsection 3.5.7 Sample Problem and Solution

Sample problem Example B.1.16.
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