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CAL - DESIG 16 - Study Guide - Midterm

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EE 16A. Designing Information Devices and Systems I Catalog Description: This course and its follow-on course EE16B focus on the fundamentals of designing modern information devices and systems that interface with the real world. Together, this course sequence provides a comprehensive foundation for core EECS topics in signal processing, learning, control, and circuit design while introducing key linear-algebraic concepts motivated by application contexts. Modeling is emphasized in a way that deepens mathematical maturity, and in both labs and homework, students will engage computationally, physically, and visually with the concepts being introduced in addition to traditional paper/pencil exercises. The courses are aimed at entering students as well as non-majors seeking a broad foundation for the field. Units: 4.0 Prerequisites: Math 1A, Math 1B or equivalent (may be taken concurrently), CS 61A or equivalent (encouraged to be taken concurrently). Credit Restrictions: Students will receive no credit for Electrical Engineering 16A after completing Electrical Engineering 20 or 40. Formats: Fall: 3.0 hours of lecture, 2.0 hours of discussion, and 3.0 hours of laboratory per week Summer: 6.0 hours of lecture, 4.0 hours of discussion, and 6.0 hours of laboratory per week Grading basis: letter Final exam status: Written final exam conducted during the scheduled final exam period Also listed as: EL ENG 16A Class Schedule (Spring 2018): TuTh 5:00PM - 6:29PM, Wheeler 150 – Grace Huilin Zhang, Hannah Dan-Feng Li, Laura Waller, Vladimir M. Stojanovic Class Schedule (Fall 2018): TuTh 9:30AM - 10:59AM, Wheeler 150 – Gireeja Ranade, Vladimir M. Stojanovic EECS 16A Designing Information Devices and Systems I Spring 2018 Lecture Notes Note 1 1.1 Introduction to Linear Algebra — the EECS Way In this note, we will teach the basics of linear algebra and relate it to the work we will see in labs and EECS in general. We will introduce the concept of vectors and matrices, show how they relate to systems of linear equations, and discuss how these systems of equations can be solved using a technique known as Gaussian elimination. 1.1.1 What Is Linear Algebra and Why Is It Important? • Linear algebra is the study of vectors and their transformations. • A lot of objects in EECS can be treated as vectors and studied with linear algebra. • Linearity is a good first-order approximation to the complicated real world. • There exist good fast algorithms to do many of these manipulations in computers. • Linear algebra concepts are an important tool for modeling the real world. As you will see in the homeworks and labs, these concepts can be used to do many interesting things in real-world-relevant application scenarios. In the previous note, we introduced the idea that all information devices and systems (1) take some piece of information from the real world, (2) convert it to the electrical domain for measurement, and then (3) process these electrical signals. Because so many efficient algorithms exist that perform linear algebraic manipulations with computers, linear algebra is often a crucial component of this processing step. 1.1.2 Tomography: Linear Algebra in Imaging Let’s start with an example of linear algebra that relates to this module and uses key concepts from this note: tomography. Tomography allows us to “see inside” of a solid object, such as the human body or even the earth, by taking images section by section with a penetrating wave, such as X-rays. CT scans in medical imaging are perhaps the most famous such example — in fact, CT stands for “computed tomography.” Let’s look at a specific toy example. A grocery store employee just had a truck load of bottles given to him. Each bottle is either empty, contains milk, or contains juice, and the bottles are packaged in boxes, with each box containing 9 bottles in a 3×3 grid. Inside a single box, it might look something like this: EECS 16A, Spring 2018, Note 1 1 If we choose symbols such that M=Milk, J=Juice, and O=Empty, we can represent the stack of bottles shown above as follows: M J O M J O M O J (1) Our grocer cannot see directly into the box, but he needs to sort them somehow (without opening every single one). However, suppose he has a device that can tell him how much light an object absorbs. This lets him shine a light at different angles to figure out how much total light the bottles in those columns absorb. Suppose milk absorbs 3 units of light, juice absorbs 2 units of light and an empty bottle absorbs 1 unit of light. If we shine light in a straight line, we can determine the amount of light absorbed as the sum of the light absorbed by each bottle. In our specific example, shining a light from left to right would look like this, where each row absorbs 6 total units of light: In order to deal with this more generally, let’s assign variables to the amount of light absorbed by each bottle: This means that x11 would be the amount of light the top left bottle absorbs, x21 would be the amount of light the middle left bottle absorbs, and so forth. Shining the light from left to right for our specific example gives the following equations: x11 +x21 +x31 = 6 (2) x12 +x22 +x32 = 6 (3) x13 +x23 +x33 = 6 (4) EECS 16A, Spring 2018, Note 1 2 Similarly, we could consider shining a light from bottom to top: Which would give the following equations: x13 +x12 +x11 = 9 (5) x23 +x22 +x21 = 5 (6) x33 +x32 +x31 = 4 (7)