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Unformatted text preview: P r e l i m i n a r y d r a f t o n l y : p l e a s e c h e c k f o r fi n a l v e r s i o n ARE211, Fall 2007 LECTURE #11: THU, OCT 4, 2007 PRINT DATE: AUGUST 21, 2007 (LINALGEBRA1) Contents 3. Linear Algebra 1 3.1. Vectors as arrows. 1 3.2. Vector operations 2 3.3. Projections 5 3.4. Proof of the cosine formula theorem 7 3.5. Linear Combinations, Linear Independence, Linear Dependence and Cones. 8 3. Linear Algebra 3.1. Vectors as arrows. Write vectors as arrows but the “real vector” is the location of the tip of the arrow. Important that in visual applications, we often draw vectors that don’t have their base at the origin. E.g., the gradient vector at x is always drawn with its base at the point x . Strictly speaking, you have to translate it back to the origin to interpret it as a vector. 1 2 LECTURE #11: THU, OCT 4, 2007 PRINT DATE: AUGUST 21, 2007 (LINALGEBRA1) 3.2. Vector operations Row and column vectors: doesn’t make any difference whether the vector is written as a row or a column vector. Purely a matter of convenience. The norm of a vector is its euclidean length: measure the arrow with a ruler. Written  x  = radicalBig ∑ n k =1 x 2 k . Note that  x  = d 2 ( x , ). Adding and subtracting vectors. Intuitive what the sum of two vectors looks like. A little trickier to figure out what the difference between two vectors looks like, but you should try to figure out the picture. How to visualize a b : do a + ( b ). Take the positive weighted sum of two vectors: α v 1 + (1 α ) v 2 . Draw it. Scalar multiples: do it. The inner product of two vectors x , y ∈ R n is the sum of the products of the components. That is, x · y = ∑ n k =1 x k y k . When I think of inner products, I think of a row vector and a column vector; purely a convention. It is hard to visualize what x · y looks like. Look at a picture of x and y and say whether x · y is positive, negative, zero. Answer is given by the angle between the two vectors. • acute angle means x · y is positive. • obtuse angle means x · y is negative. • ninety degree angle means x · y is zero. ARE211, Fall 2007 3 x a b c Figure 1. Inner Products. Theorem: a · u =  a  u  cos ( θ ), where θ is the angle between a and u . (We’ll prove this later in the lecture.) Note the beauty of cos: doesn’t matter whether you look at the big angle between the vectors or the little one, get the same answer! In Fig. 1, rank the inner products x · a , x · b and x · c . Answer: all the vectors are the same length, so that the only thing that determines the inner product is the angle between them. Hence x · a > x · b > x · c . Application: a fact that we’ll learn soon is that for small vectors dx , f ( x + dx ) ≈ f ( x )+ ∇ f ( x ) · dx ....
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 Fall '07
 Simon
 Linear Algebra, Vectors, Vector Space, OPPO, nonnegative cone

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