This optional homework assignment develops the notion of vector spaces and linear maps. It is worth the equivalent of 25 points applied to the first exam. Work must be shown to receive full credit. Correct answers without shown work will receive at most half credit. Recall that a linear function is one of the form f(x) = mr +b, where i and b are real number constants. Although this definition is satisfactory for our relatively straight- forward applications (pun intended), this exercise will develop a more general definition of linearity. A vector space is a collection of ordered lists of numbers that is characterized by the type of numbers one can choose from (called the scalar field) and the number of numbers in each list (called the dimension)- For this exercise, we will restrict our scalar field to the real numbers R. Each list-member of a vector space is called a vector. To distinguish vectors from scalars we use an arrow above the letter (e.g. #) to indicate a vector, and normal font with parentheses and commas (e.g. (1, v2, va)) to indicate its scalar components in each dimension. Vector spaces appear in many places in pure and applied mathematics. The ry-plane we are familiar with can be viewed as a vector space of dimension 2, since it consists of ordered pairs of real numbers (I, y). Engineers often use a four-dimensional vector space with points (t. r, y, 2) to describe how situations in 3d space evolve over time. Advanced string theory calls for 11 dimensions to exist at the quantum level, and some quantitative economists even consider the stock market as a very high-dimensional vector space where each stock ticker has its own dimension. In pure mathematics, sometimes special vector spaces with infinite dimensions are considered! Vector spaces have two fundamental operations. Let # and w be vectors in a space of dimension n and let c be a real number scalar. Vector addition is defined component-wise by adding each coordinate separately: 7 + w = (v1, "2, va, . .., Un) + (w1, w2, wa, . .., Wn) = (vi + w1, "2 + w2, us + wg, ..., Un + Wn). Scalar multiplication is defined component-wise by multiplying each coordinate by the scalar: c = c(v1, U2, "3; . .- ; "n) = (CV1, co2, cuz, . . .; CUn) The conventional way to describe the vector space as a whole is to indicate its field and then include a superscript (like an exponent) for the dimension (e.g. R for three dimensions). In the theory of vector spaces, an important set of functions are those from a space R" to itself. Since writing coordinates can quickly become tedious in high dimensions, we use the shorthand f : R" - Rf : R" - R to indicate that the function f takes vectors (lists of n numbers) from R" as inputs and gives vectors from R" as outputs. For vector spaces, a function f is said to be a linear map if and only if two conditions are satisfied for any two vectors f, w inside R" and for any scalar c inside R: f(3+ 0) = f(0)+f(@) and f(c) = of (3). The remainder of this assignment is designed to give you some experience with linear maps of this form. Part i. (5 points): Explain why the definition of a linear function does NOT satisfy the definition of a linear map considered as f : R] = R'.Part 1. (5 points): The natural projection onto the kth coordinate s a function a tR = R" detined by mg(v)] = (U, .. 0w k. 0). In other words, it sets all but the kth component of v equal to U while keeping the kth component the same. Show that mp satisfies the conditions for a linear map. We will now consider another special function from R' to itself, the orthogonal projection onto a line 6, written: proje : R' - R'. This function can be thought of as taking a vector # = (21, 1/1) and sending it to its "shadow" on the line / with a "light" that is perpendicular to f above 7 as pictured below. proj,() Part iii. (15 points): Let 7 = (21, yi) as above, and suppose that / has the equation y = mr. Determine a formula for the coordinates of proj,(7) in terms of 21, y1, and m. Then determine whether proje satisfies the conditions for a linear map