Chapter 1.1, Problem 1SP

A savings account is started with an initial deposit of $500. The account earns 1.5% interest compounded annually. (a) Write an equation to represent the amount of money in the account as a function of time in years. (b) Find the amount of time it takes for the account balance to reach $800. Show your work.

(a) Use the fundamental theorem of algebra to determine the number of roots for 2x² +4x+7. (b) What are the roots of 2x² +4x+7? Show your work.

Consider the function f(x)=x³ + 2x² − 3 (a) Graph the function. (b) What are the x- and y-intercepts of the graph?

Let 2 A = 4 3 -4 0 1 (a) Show that v = eigenvalue. () is an eigenvector of A and find the corresponding (b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a) may be useful. (c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces. (d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.

(c) Let 6 0 0 A = -10 4 8 5 1 2 (i) Find the characteristic polynomial of A and factorise it. (ii) Determine all eigenvalues of A and find bases for the corresponding eigenspaces. (iii) Is A diagonalisable? Give reasons for your answer.

most 2, and let Let P2 denote the vector space of polynomials of degree at D: P2➡ P2 be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative p'(t) 2at+b, that is, D(p) = p'. (a) Prove that D is a linear transformation. (b) Find a basis for the kernel ker(D) of the linear transformation D and compute its nullity. (c) Find a basis for the image im(D) of the linear transformation D and compute its rank. (d) Verify that the Rank-Nullity Theorem holds for the linear transformation D. (e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.

(c) Let A = -1 3 -4 12 3 3 -9 (i) Find bases for row(A), col(A) and N(A). (ii) Determine the rank and nullity of A, and verify that the Rank-Nullity Theorem holds for the above matrix A.

-(0)-(0)-(0) X1 = x2 = x3 = 1 (a) Show that the vectors X1, X2, X3 form a basis for R³. y= (b) Find the coordinate vector [y] B of y in the basis B = (x1, x2, x3).

Let A 1 - 13 (1³ ³) 3). (i) Compute A2, A3, A4. (ii) Show that A is invertible and find A-¹.