\(\mathbb{R}^n\) as a set of points
We write \(\mathbb{R}^n\) to denote the set of all points in \(n\)-dimensional space.
A point is represented by an ordered \(n\)-tuple of real numbers \((x_1, x_2, \dots, x_n)\).
The numbers \(x_1, x_2, \dots, x_n\) are called the coordinates of the point.
\(\mathbb{R}^n\) as a set of vectors
We write \(\mathbb{R}^n\) to denote the set of all vectors in \(n\)-dimensional space.
Such a vector is represented as a \(n\)-dimensional column vectors written as \[ \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \] where \(x_1, x_2, \dots, x_n\) are real numbers.
The numbers \(x_1, x_2, \dots, x_n\) are called the components of the vector.
All the definitions and properties of vector algebra, norms, and dot products that we discussed for \(\mathbb{R}^2\) and \(\mathbb{R}^3\) also carry over to \(\mathbb{R}^n\) with only minor modifications.
For exmaple, given two vectors \[\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{bmatrix} \quad \text{and} \quad \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix},\] their sum/difference is defined as \[\mathbf{u} \pm \mathbf{v} = \begin{bmatrix} u_1 \pm v_1 \\ u_2 \pm v_2 \\ \vdots \\ u_n \pm v_n \end{bmatrix}.\]
Their norm is defined as \[\lVert \mathbf{u} \rVert = \sqrt{u_1^2 + u_2^2 + \dots + u_n^2}.\]
Their inner product is defined as \[\langle \mathbf{u}, \mathbf{v} \rangle = u_1v_1 + u_2v_2 + \dots + u_nv_n.\]
Standard basis vectors in \(\mathbb{R}^n\)
The standard basis vectors in \(\mathbb{R}^n\) are denoted by \(\mathbf{e}_1\), \(\mathbf{e}_2\), \(\dots\), \(\mathbf{e}_n\), and are defined as \[ \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \quad \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \quad \mathbf{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \quad \dots, \quad \mathbf{e}_n = \begin{bmatrix} 0 \\ 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}. \]
Theorem: Resolving vectors into standard basis vectors
Any vector
\[\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{bmatrix}\]
in \(\mathbb{R}^n\) can be expressed as a linear combination of the standard basis vectors (i.e., a sum of scalar multiples of the basis vectors) as:
\[\mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + \dots + u_n \mathbf{e}_n.\]
a. Practice with vector algebra
Let
\[ \mathbf{u} = \begin{bmatrix} 1 \\ 2 \\ -1 \\ 3 \end{bmatrix} \quad \text{and} \quad \mathbf{v} = \begin{bmatrix} 0 \\ -1 \\ 2 \\ 1 \end{bmatrix} \]
b. Practice with inner produts and norms
With the same vectors \(\mathbf{u}\) and \(\mathbf{v}\) as in part (a):
Matrices
An \(m\times n\) matrix is a rectangular array of numbers with \(m\) rows and \(n\) columns, typically written as \[ A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix}. \]
Matrix algebra
Matrix addition and subtraction is performed element-wise. However, both matrix must have the same number of rows and columns.
Scalar multiplication is performed by multiplying each entry of the matrix by the scalar.
Matrix multiplication is defined when the number of columns of the first matrix equals the number of rows of the second matrix. The product is obtained by taking the dot product of rows and columns. (See the next exercise.)
a. Practice with matrix addition/subtraction/scaling
Let
\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & -1 \\ 2 & 1 \\ -3 & 0 \end{bmatrix}. \]
b. Practice with matrix multiplication
Let
\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}, \quad C = \begin{bmatrix} 0 & -1 \\ 2 & 1 \end{bmatrix}. \]
Matrices multiplied by vectors
Let
\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}. \]
Important to remember: A function \(f\) is a “machine” that accepts an input, and produces an output.
The set \(D\) of inputs is called the domain of \(f\), while the set \(C\) that contains the outputs is called the codomain.
The actual set of outputs is called the range of \(f\).
This is represented notationally as \(f:D\to C\).
A familiar function from pre-calculus
Suppose you’re in a pre-calculus class (or calculus, I guess) and your instructor writes \(f(x) = x^2\) on the board.
Now do the same four exercises, but with the expression \(g(x) = \sqrt{x}\).
Theorem: Every matrix yields a function
Let \(A\) be an \(m\times n\) matrix. Then \(A\) defines a function \(T:\mathbb{R}^n \to \mathbb{R}^m\) by the formula
\[ T(\mathbf{v}) = A\mathbf{v}. \]
In words, \(T\) is a function that takes an \(n\)-dimensional vector and outputs an \(m\)-dimensional vector.
Note
Let
\[ A = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} \]
Then \(A\) defines a function \(T\), as described on the previous slide.
Determinants of \(2\times 2\) matrices
Let
\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}. \]
The determinant of \(A\) is defined as the number
\[ \det(A) = ad - bc. \]
Theorem: \(2\times 2\) determinants and area
Let
\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}. \]
The area of the parallelogram with sides given by the columns of \(A\) is \(|\det(A)|\).
a. Computing \(2\times 2\) determinants
Compute the determinants of the following matrices.
\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \]
\[ B = \begin{bmatrix} 0 & -1 \\ 2 & 3 \end{bmatrix} \]
\[ C = \begin{bmatrix} 1 & 0 \\ -1 & 0 \end{bmatrix} \]
b. Areas of parallelograms
Compute the areas of the parallelograms whose sets of vertices contain the following points:
Determinants of \(3\times 3\) matrices
Let
\[ A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}. \]
The determinant of \(A\) is defined as the number
\[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg). \]
Theorem: \(3\times 3\) determinants and volume
Let
\[ A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}. \]
The volume of the parallelepiped with sides given by the columns of \(A\) is \(|\det(A)|\).
a. Computing \(3\times 3\) determinants
Compute the determinants of the following matrices.
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
\[ B = \begin{bmatrix} 0 & -1 & 2 \\ 2 & 3 & 1 \\ 1 & 0 & -1 \end{bmatrix} \]
\[ C = \begin{bmatrix} 1 & 0 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
b. Volumes of parallelepipeds
Compute the volumes of the parallelepipeds whose sets of vertices contain the following points: