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====== Math 445 lecture 2: Vectors ====== topics <code> [] % creating vector vectors with square brackets () % accessing/assigning vector elements with parentheses ' % the transpose operator, apostrophe : % the colon operator, making range vectors + - % vector addition, subtraction .* ./ .^ % elementwise vector operations </code> ===== Vectors in math ===== Vectors are hugely important in math, science, engineering. In math, a vector is a list of numbers. E.g. this is a 3-d vector <latex> x = \left(\begin{array}{c} 4.3 \\ 5.9 \\ 0.1 \end{array} \right) </latex> In math, the components or elements of a vector are written with subscripts. E.g. for the above vector <latex> x_1 = 4.3, \quad x_2 = 5.9, \quad x_3 = 0.1 </latex> ===== Creating vectors with explicit lists in square brackets ===== In Matlab, there's a distinction between row vector and column vectors. You can construct **row vectors** by listing elements between **square brackets**, with either **spaces** or **commas** between. <code> >> x = [4.3 5.9 0.1] % assign a row vector into variable x x = 4.3000 5.9000 0.1000 </code> To create a column vectors, put semicolons between the elements. <code> >> x = [4.3; 5.9; 0.1] % assign a col vector into variable v x = 4.3000 5.9000 0.1000 </code> ===== Accessing elements with parentheses ===== In Matlab, you access the elements of a vector using **parentheses**. E.g. given the above vector $x$, you **get** the 2nd component of $x$ this way <code> >> x(2) ans = 5.9000 </code> Note what happened there. We asked Matlab for the 2nd element of x. It **returned** the value 5.9 and assigned it to the default return variable ''ans''. Alternatively, the following will **assign** or **set** the value of the second component of $x$ to another number. <code> >> x(2) = 2.7 x = 4.3000 2.7000 0.1000 </code> ===== The transpose operator ' (apostrophe) ===== Note that last method. In Matlab, the apostrophe '' ' '' stands for the **transpose**. The transpose operator turns a row vector into a column vector, and vice versa. <code> >> x = [4.3 5.9 0.1] x = 4.3000 5.9000 0.1000 >> x' ans = 4.3000 5.9000 0.1000 >> y = x' y = 4.3000 5.9000 0.1000 >> z = y' z = 4.3000 5.9000 0.1000 </code> ===== Constructing vectors with the colon operator : ===== The Matlab **colon** operator : is used in a number of ways. Here we'll see how it can be used to create vectors. The most straightforward of these is the syntax ''m:n'', which creates a row vector of the integers from ''m'' to ''n''. E.g. <code> >> 1:10 ans = 1 2 3 4 5 6 7 8 9 10 >> x = 4:7 x = 4 5 6 7 </code> You can also specify a **step** or **increment** between the elements using the syntax ''m:step:n''. For example, ''5:2:17'' will produce a vector of odd integers between 5 and 17. <code> >> x = 5:2:17 x = 5 7 9 11 13 15 17 </code> Lastly, the colon operator works with nonintegers, too. E.g. <code> >> x = 0:0.1:0.7 x = 0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 </code> ===== Constructing vectors with linspace : ===== In Matlab, vectors are often used to represent *discretized functions*, that is, the values of a function at a discrete set of gridpoints ===== Supplemental material ===== [[https://www.youtube.com/watch?v=zXP_pr7np-o | ' (apostrophe), Frank Zappa (the song)]] [[https://www.youtube.com/watch?v=kQDp2425WQ0&list=PL8WvZFiJpAr29HFJkRcE4NvWCeYy3ytte | ' (apostrophe), Frank Zappa (full album) ]]
