In Problems 17–20 the given
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- 3. Let a ER and let: 1 -2 2 4- ( -² ) A = 1+ a 4+ a -2. a " a a 2 - a You may assume without calculation that ch₁(x) = (x − 2)²(x − 3). For which values of a, is A diagonalisable?arrow_forwardProblem 25. Suppose yı (x), y2(x), and y3(x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 for each case.arrow_forwardProblem 16 (#2.3.34).Let f(x) = ax +b, and g(x) = cx +d. Find a condition on the constants a, b, c, d such that f◦g=g◦f. Proof. By definition, f◦g(x) = a(cx +d) + b=acx +ad +b, and g◦f(x) = c(ax +b) + d=acx +bc +d. Setting the two equal, we see acx +ad +b=acx +bc +d ad +b=bc +d (a−1)d=(c−1)b That last step was merely added for aesthetic reasons.arrow_forward
- Part 3&4 needed to be solved correctlyarrow_forward7. Problem 8.3.14: Determine whether the given set of vectors is linearly dependent or linearly independent. U1 = (2,1,1,5), ūz = (2,2,1,1), ūz = (3,–1,6,1), ū4 = (1,1,1,–1)arrow_forward14. Assume x E R. Give the matrix associated with the quadratic form 10(x,) - 5x,x2 + 6(x,).arrow_forward
- 1.5.17 Suppose the solution set of a certain system of linear equations can be described as x, = 5+. 4x3, X2 = - 6 – 7x3, with x3 free. Use vectors to describe this set as a line in R°. %D Geometrically, the solution set is a line through || parallel toarrow_forwardShow mathematically that (10-6) is not independent of (10-1),(10-3), and (10-4).arrow_forward3. Suppose A = 3 1 1 -2 (a) Find a vector X E R² such that AX = (b) Find a vector X € R² such that AX = B 9arrow_forward
- Question 9 Find all the roots of z3 – 3(5 +j) = 0 and give the answers in rectangular form. Question 10 Use Crammer's rule to solve the following linear system for y only. 2x – 3y = 3 – z 4x +y = -4 = 3y + z-2 İLIFE Digitalarrow_forward1 1. Find numbers x, y and z such that the matrix A= 1 0 4 -2 = I3 satisfies 1) 4arrow_forward1. The vector X = k is a solution of x - (: -)x-G) 8, X' 2 for k = Enter only the value of k.arrow_forward
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