(b) In each part, determine whether the given vector 2x - x2 +x + 3 € P3 (R) is in the span of S = {x³ + x² + x + 1, x2 + x + 1, x + 1}. -x³ + 2x2 + 3x +3 € P3 (R) is in the span of S = {x³ + x² + x + 1, x2 + x + 1, x +1}. %3D i. ii.

Q: Find two positive real numbers such that they add to 40 and their product is as large as possible.

A: Our Aim is to find two positive real numbers a and b such that they add to 40 and their product is…

Q: Use Cramer's rule (by hand) to find the unique solution of x1 +3x3 = -1 -3x1 +2x3 = 1 2x2-x3=0

A:

Q: Show that the square of any integer is of the form 3m or 3m+1 but not of form 3m+2.

A: Proof :-

Q: = (x+ )². 12 2²2 +6x +. = (x+ 2² + 6x +9= (x+3) ².

A:

Q: Determine the real values of a, b, and c for the quadratic function Ax) = ax + bx+c, a +0 that…

A: Here we have to determine the real values of a, b, and c for the quadratic function f(x)=ax2+bx+c…

Q: Find the curl of x²zi+0j+xz²k.

A:

Q: Show that Q(4 - i) = Q(1 + i), where i = √-1.

A: To show that ℚ4-i=ℚ1+i. That is we need to show that ℚ4-i⊆ℚ1+i and ℚ1+i⊆ℚ4-i. Here i=-1 that is…

Q: Let p, = 2t – 1 and p2 = t3 +2. Write three polynomials that are in span{p1, P2}-

A:

Q: Let A 1 2 −1 -2 3 -2 -1 1 1 Diagonalize A to find A100.

A:

Q: Find matrix X:   3.     6     5     4     -2   *X=     18     49     -20     6

A: The detailed solution is given below Thank you

Q: Calculate the area of the parallelogram determined by the points (-1 , -1), (2 , 4), (4 , 3) and (7…

A:

Q: Explain why v = ( 1, 0, -1) cannot be a row of A and also in the nullspace.

A:

Q: Prove that (p – q)× (p+q)= 2(p×q)

A:

Q: Find p so that  x2 - 6x + p = 0  has one root (i.e. is a perfect square)

A: We will solve it by two methods Shortcut method to find p for the equation given is x2 + b x + c  is…

Q: Use the

A: we have; 3x1-2x2+4x3=213x1+4x2-2x3=92x1-x2-x3=10

Q: Prove that 3n² - 1 is never a perfect square.

A: Perfect square: An integer n is said to be a perfect square if n=k.k for some integer k  We now…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.