Consider the following applications of properites of the real numbers \(\mathbb R\text{:}\)

1. \(1+(2+3)=(1+2)+3\text{.}\)

2. \(7+4=4+7\text{.}\)

3. There exists some \(\unknown\) where \(5+\unknown=5\text{.}\)

4. There exists some \(\unknown\) where \(9+\unknown=0\text{.}\)

5. \(\frac{1}{2}(1+7)\) is the only number that is equally distant from \(1\) and \(7\text{.}\)

Which of the following properites of \(\IR^2\) Euclidean vectors is NOT true?

1. \(\left[\begin{array}{c} x_1\\x_2\end{array}\right] +\left(\left[\begin{array}{c} y_1\\y_2\end{array}\right] +\left[\begin{array}{c} z_1\\z_2\end{array}\right]\right)= \left(\left[\begin{array}{c} x_1\\x_2\end{array}\right] +\left[\begin{array}{c} y_1\\y_2\end{array}\right]\right) +\left[\begin{array}{c} z_1\\z_2\end{array}\right]\text{.}\)

2. \(\left[\begin{array}{c}x_1\\x_2\end{array}\right] + \left[\begin{array}{c}y_1\\y_2\end{array}\right] = \left[\begin{array}{c}y_1\\y_2\end{array}\right] + \left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)

3. There exists some \(\left[\begin{array}{c}\unknown\\\unknown\end{array}\right]\) where \(\left[\begin{array}{c}x_1\\x_2\end{array}\right] +\left[\begin{array}{c}\unknown\\\unknown\end{array}\right] =\left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)

4. There exists some \(\left[\begin{array}{c}\unknown\\\unknown\end{array}\right]\) where \(\left[\begin{array}{c}x_1\\x_2\end{array}\right]+ \left[\begin{array}{c}\unknown\\\unknown\end{array}\right]= \left[\begin{array}{c}0\\0\end{array}\right]\text{.}\)

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 \(\left[\begin{array}{c}y_1\\y_2\end{array}\right]\text{.}\)

Consider the following applications of properites of the real numbers \(\mathbb R\text{:}\)

1. \(3(2(7))=(3\cdot 2)(7)\text{.}\)

2. \(1(19)=19\text{.}\)

3. There exists some \(\unknown\) such that \(\unknown \cdot 4= 9\text{.}\)

4. \(3\cdot (2+8)=3\cdot 2+3\cdot 8\text{.}\)

5. \((2+7)\cdot 4=2\cdot 4+7\cdot 4\text{.}\)

Which of the following properites of \(\IR^2\) Euclidean vectors is NOT true?

1. \(a\left(b\left[\begin{array}{c}x_1\\x_2\end{array}\right]\right)= ab\left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)

2. \(1\left[\begin{array}{c}x_1\\x_2\end{array}\right]= \left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)

3. There exists some \(\unknown\) such that \(\unknown\left[\begin{array}{c}x_1\\x_2\end{array}\right]= \left[\begin{array}{c}y_1\\y_2\end{array}\right]\text{.}\)

4. \(a(\vec u+\vec v)=a\vec u+a\vec v\text{.}\)

5. \((a+b)\vec v=a\vec v+b\vec v\text{.}\)

Every Euclidean vector space \(\mathbb R^n\) satisfies the following properties, where \(\vec u,\vec v,\vec w\) are Euclidean vectors and \(a,b\) are scalars.

1. Vector addition is associative: \(\vec u + (\vec v + \vec w)= (\vec u + \vec v) + \vec w\text{.}\)

2. Vector addition is commutative: \(\vec u + \vec v= \vec v + \vec u\text{.}\)

3. An additive identity exists: There exists some \(\vec z\) where \(\vec v + \vec z=\vec v\text{.}\)

4. Additive inverses exist: There exists some \(-\vec v\) where \(\vec v + (-\vec v)=\vec z\text{.}\)

5. Scalar multiplication is associative: \(a (b \vec v)=(ab) \vec v\text{.}\)

6. 1 is a multiplicative identity: \(1 \vec v=\vec v\text{.}\)

7. Scalar multiplication distributes over vector addition: \(a (\vec u + \vec v)=(a \vec u) + (a \vec v)\text{.}\)

8. Scalar multiplication distributes over scalar addition: \((a+ b) \vec v=(a \vec v) + (b \vec v)\text{.}\)

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.

1. Vector addition is associative: \(\vec u\oplus (\vec v\oplus \vec w)= (\vec u\oplus \vec v)\oplus \vec w\text{.}\)

2. Vector addition is commutative: \(\vec u\oplus \vec v= \vec v\oplus \vec u\text{.}\)

3. An additive identity exists: There exists some \(\vec z\) where \(\vec v\oplus \vec z=\vec v\text{.}\)

4. Additive inverses exist: There exists some \(-\vec v\) where \(\vec v\oplus (-\vec v)=\vec z\text{.}\)

5. Scalar multiplication is associative: \(a\odot(b\odot\vec v)=(ab)\odot\vec v\text{.}\)

6. 1 is a multiplicative identity: \(1\odot\vec v=\vec v\text{.}\)

7. Scalar multiplication distributes over vector addition: \(a\odot(\vec u\oplus \vec v)=(a\odot\vec u)\oplus(a\odot\vec v)\text{.}\)

8. Scalar multiplication distributes over scalar addition: \((a+ b)\odot\vec v=(a\odot\vec v)\oplus(b\odot \vec v)\text{.}\)

Consider the set \(\IC\) of complex numbers with the usual defintion for addition: \((a+b\mathbf i)\oplus(c+d\mathbf i)=(a+c)+(b+d)\mathbf i\text{.}\)

Let \(\vec u=a+b\mathbf{i}\text{,}\) \(\vec v=c+d\mathbf{i}\text{,}\) and \(\vec w=e+f\mathbf{i}\text{.}\) Then

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 non-Euclidean vector space.

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.

• \(\IC\text{:}\) Complex numbers.

• \(M_{m,n}\text{:}\) Matrices of real numbers with \(m\) rows and \(n\) columns.

• \(\P_n\text{:}\) Polynomials of degree \(n\) or less.

• \(\P\text{:}\) Polynomials of any degree.

• \(C(\IR)\text{:}\) Real-valued continuous functions.

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\text{.}\)

Which of the following vectors is not in \(V\text{?}\)

1. \(\displaystyle (0, 0)\)

2. \(\displaystyle (1, 2)\)

3. \(\displaystyle (2, 4)\)

4. \(\displaystyle (3, 8)\)

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with the operation \(\oplus\) defined by

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 \(\vec u\oplus\vec v\text{?}\)

1. \(\displaystyle (2, 6)\)

2. \(\displaystyle (2, 8)\)

3. \(\displaystyle (3, 6)\)

4. \(\displaystyle (3, 8)\)

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by

Let \(a=2, b=-3\) be scalars and \(\vec u=(1,2) \in V\text{.}\)

Part 1.

Verify that

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by

Let \(a=2, b=-3\) be scalars and \(\vec u=(1,2) \in V\text{.}\)

Part 2.

Compute the value of

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by

Let \(a, b\) be unspecified scalars in \(\mathbb R\) and \(\vec u = (x,y)\) be an unspecified vector in \(V\text{.}\)

Part 1.

Show that both sides of the equation

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by

Let \(a, b\) be unspecified scalars in \(\mathbb R\) and \(\vec u = (x,y)\) be an unspecified vector in \(V\text{.}\)

Part 2.

Show that \(V\) contains an additive identity element \(\vec{z}=(\unknown,\unknown)\) satisfying

That is, pick appropriate values for \(\vec{z}=(\unknown,\unknown)\) and then simplify \((x,y)\oplus(\unknown,\unknown)\) into just \((x,y)\text{.}\)

Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by

Let \(a, b\) be unspecified scalars in \(\mathbb R\) and \(\vec u = (x,y)\) be an unspecified vector in \(V\text{.}\)

Part 3.

Is \(V\) a vector space?

1. Yes

2. No

3. More work is required

It turns out \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by

Thus, \(V\) is a vector space.

Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by

Part 1.

Show that \(1\) is the scalar multiplication identity element by simplifying \(1\odot(x,y)\) to \((x,y)\text{.}\)

Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by

Part 2.

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.

Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by

Part 3.

Is \(V\) a vector space?

1. Yes

2. No

3. More work is required

Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by

Part 1.

Show that scalar multiplication distributes over vector addition, i.e.

Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by

Part 2.

Show that vector addition is not associative, i.e.

Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by

Part 3.

Is \(V\) a vector space?

1. Yes

2. No

3. More work is required