Suppose one has a binary asymmetric channel K: S→ R represented by and Pris(r|1)-(-1)+(-0) = = = 1/3 2/3 1/2

Let A 1 E 1 = 2 — 0 3 Compute: (ATA)²

Consider the following interpolating points for the function f(x) = x² – 2x: Xi 01 23 f(xi) 0 -1 03 (a) Use monomial basis to find P₁(x) that interpolates f(x) at x; for i = 0,1. Con- struct the Vandermonde matrix and then use the backslash \ in matlab to solve the linear system. (b) Repeat (a) to find P₂(x) that interpolates f(x) at x; for i = 0, 1, 2. (c) Repeat (a) to find P3(x) that interpolates f(x) at x, for i = 0, 1, 2, 3. (d) Compare P₂(x), P3(x) and f(x). What do you see? Can you explain it?

Let T be a linear transformation from M2,2 into M2,2 such that 1 -1 02 12 0 2 0 1 ¹([² :) - [ ²² ([:)) - [³ ] ([ ² ) = [: ²] -([¦ :)) = [1 ²] ] 3-1 1 3 Find ([²1]) 4 -1

6. The covariance between X and Y is calculated and recorded as SXY (and sxy is a negative number). A linear transformation is performed on X. The linearly transformed variable is called W, where W; = 5 + Xi. The covariance between W and Y is calculated and recorded as sWY. Which statement is correct? A) В) In this example, swy would be less than sxY. In this example, swY Would be equal to sxY. In this example, swy would be greater than SXXY. 7. The covariance between X and Y is calculated and recorded as sxy (and sxy is a positive number). A linear transformation is performed on X. The linearly transformed variable is called W, where W; = (4)X;. The covariance between W and Y is calculated and recorded as sWY. Which statement is correct? In this example, sxY would be less than swy. In this example, sxY Would be equal to swy. In this example, sxY would be greater than sWY. A) B) 8. The correlation between X and Y is calculated and recorded as rxY (and rxy is positive). A linear…

section 2.2: Number 4

4.3 Compute the eigenspaces of а. A:= b. -2 2]

Let T be a linear transformation from M2,2 into M2,2 such that ([:))-[2] ([:))-[:] [:))-[:] [:))-[1] Find ([-1³). 3 12 7 -1 3 ↓ 1 X Need Help? Read It Watch It MY N

[:] Suppose that T is a linear transformation from ³ into ² such that 3 -[-2]. (₂) - [8].a T -[2], and T(3) = [1] The columns of k T(e₁) O *1 Tx2 = 2x1 x3 *1 TX2 002 1 0 0 0 Tx2 = *3 3x1-2x2] Tx2 = 2x1 001 Find a formula for the image of an arbitrary x= x2 in ³. 4x2 + x3 1 0 are e₁ 3x1 + 2x2 - 4x3 -2x1 + x3 3x1-2x1 2x1 3x1 + 2x2-4x3 -2x1 wwww + X3 x3 0 e2 ез 0 akan wwwwww.

17.2.4 (a) Once you have a matrix representation of any group, a 1-D representation can be obtained by taking the determinants of the matrices. Show that the multiplicative relations are preserved in this determinant representation. (b) Use determinants to obtain a 1-D representation of D3 from the 2 x 2 representa- tion in Eq. (17.2).