(a) Let A, B be n × n symmetric matrices such that XTAX = X™BX for all column vectors X e R". Show that A = B.

8) Let A be an nxn ma triX. vector b the equation Ax = b has more than one solution. Explain why A is not an invertible Matrix.

If A and B are invertible matrices and c is any scalar, then which of the following does not hold? (A-1)-1 = A (A")-1 = (A¯1)" (cA)~1 = {A-1 (AB)¯1 = A-1B-1

Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 3 -1 - -6 15 2-5 x=x2+x3 (Type an integer or fraction for each matrix element.)

The set of non-zero vectors {b,, b,, bz, b4} are vectors in R* with the property that 4b, + 2b, = 6b, . Let matrix B = | b, b2 b; b4. (a) What does the matrix b, b, b, reduce to in RREF form? (b) Find a specific solution to the matrix equation Bx = 0. That is, find an x vector that solves that equation. (c) What is the maximum number of pivots matrix B can have? Please explain briefly referencing anything from above. (d) Can the set of vectors {b1, b2, b3, b4} span R’? Explain why or why not? (e) Can the set of vectors {b,, b2, b3, b4} span R*? Explain why or why not? (f) Matrix M is 7 x 4. If possible, show that the columns of the matrix MB are linearly dependent. If it is not possible to show this, explain why.

3= b, ,b2} and C= (c,.c2} be bases for a vector space V, and suppose b, = - 6c, + 5c2 and b2 = - 4c, +2c2. a. Find the change-of-coordinates matrix from B to C. b. Find (x]c for x = - 5b, + 3b2. Use part (a). a. P = C-B b. (xlc (Simplify your answers.)