Summation notation
- Expand and evaluate: \[ \sum_{i=1}^{4}(2i+1) \]
\[ \sum_{i=1}^{4}(2i+1) = 3+5+7+9 = 24 \]
- Evaluate: \[ \sum_{i=1}^{6}i^3 \]
\[ \sum_{i=1}^{6}i^3 = 1^3+2^3+3^3+4^3+5^3+6^3 = 1+8+27+64+125+216 = 441 \]
- A researcher records daily rainfall (in mm) over 5 days: \(x=(2,0,5,3,4)\). Find the sample mean, given by \[ \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i. \]
\[ \bar{x}=\frac{1}{5}(2+0+5+3+4) = \frac{14}{5} = 2.8\text{ mm} \]
- A set of tree heights (in m) is \(x=(2,4,6,8)\), with sample mean \(\bar{x}=5\). Find the sample variance, given by \[ s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2. \]
\[ s^2=\frac{1}{4-1}\left[(2-5)^2+(4-5)^2+(6-5)^2+(8-5)^2\right] = \frac{1}{3}(9+1+1+9) = \frac{20}{3} \approx 6.67\text{ m}^2 \]
Vectors
- Given \(r=(-3,6)\) and \(s=(5,2)\), find \(r+s\) and \(r-s\).
\[ r+s=(-3+5,\ 6+2)=(2,8) \] \[ r-s=(-3-5,\ 6-2)=(-8,4) \]
- Given \(u=(2,-3)\), find \(w=-2u\). What effect does scalar multiplication have on a vector?
\[ w=-2u=(-4,6) \] Scalar multiplication changes the vector’s magnitude (length) but not its direction — unless the scalar is negative, in which case the direction reverses (as it does here).
- Find the dot product of \(p=(4,0,-3)\) and \(q=(-2,5,1)\)
\[ p\cdot q = (4)(-2)+(0)(5)+(-3)(1) = -8+0-3 = -11 \]
- Are the vectors \(m=(5,-2)\) and \(n=(4,10)\) orthogonal?
\[ m\cdot n = (5)(4)+(-2)(10) = 20-20 = 0 \] Since the dot product is 0, \(m\) and \(n\) are orthogonal.
Matrices
- A matrix \(B\) has dimensions \(5\times 2\). How many rows and columns does it have, and what does \(b_{41}\) refer to?
\(B\) has 5 rows and 2 columns. \(b_{41}\) refers to the element in row 4, column 1.
- Given \(A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\) and \(B=\begin{bmatrix}5&-1\\0&2\end{bmatrix}\), find \(A+B\).
\[ A+B=\begin{bmatrix}1+5&2-1\\3+0&4+2\end{bmatrix}=\begin{bmatrix}6&1\\3&6\end{bmatrix} \]
- Can you add matrices of different sizes?
No. Matrix addition is defined element-by-element, so the two matrices must have the same dimensions.
- Given \(A=\begin{bmatrix}2&0\\1&3\end{bmatrix}\), find \(4A\).
\[ 4A=\begin{bmatrix}8&0\\4&12\end{bmatrix} \]
- Multiply the matrices: \(\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}5&6\\7&8\end{bmatrix}\)
Each entry is the dot product of a row of the first matrix and a column of the second: \[ \begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}5&6\\7&8\end{bmatrix}=\begin{bmatrix}(1)(5)+(2)(7)&(1)(6)+(2)(8)\\(3)(5)+(4)(7)&(3)(6)+(4)(8)\end{bmatrix}=\begin{bmatrix}19&22\\43&50\end{bmatrix} \]
- If you multiply a \(2\times 7\) matrix by a \(7\times 4\) matrix, what are the dimensions of the resulting matrix?
The result is a \(2\times 4\) matrix — the number of rows of the first matrix by the number of columns of the second.
- Two \(2\times 2\) rasters represent land surface temperature (°C) in 2010 (\(T_{2010}\)) and 2020 (\(T_{2020}\)) across four grid cells. Find the raster of temperature change, \(\Delta T = T_{2020}-T_{2010}\). Given \[ T_{2010}=\begin{bmatrix}22&24\\19&21\end{bmatrix}, \qquad T_{2020}=\begin{bmatrix}25&28\\20&23\end{bmatrix} \] find \(\Delta T\).
\[ \Delta T = T_{2020}-T_{2010} = \begin{bmatrix}25-22&28-24\\20-19&23-21\end{bmatrix} = \begin{bmatrix}3&4\\1&2\end{bmatrix} \]