Linear Algebra Answers

Given any n∈N , it is possible to define a linear transformation whose kernel has dimension n.

True or false with full explanation

The function < , > : R^2× R^2→ R : <(x1), (y1)>

<(X2),(y2)>= x1^2+x2^2 define inner product over R^2

True or false with full explanation

If V={(x,y)∈R^2 | x=y} , then (1,0) +V and (0,1)+V are two distinct element of R^2/V.

True or false with full explanation

Check whether or not the set of all symmetric matrix in Mn(R) form a real vector space with respect to the usual addition and scalar multiplication for Mn(R)

Let T: R^2→R^2 be a linear opeator with matrix

[ 7 1]

[-1 1] (w.r.t. the standard basis). Use Cayley haMilton theorem to check whether T is invertible or not. If T is invertible, obtain T^-1(x,y) for (x,y)∈ R^2. If T is not invertible, obtain the minimal polynomial of T.

Is {sinx, cosx, sin(x+π/4)} a linearly independent set over R ? Justify our answer

Apply the fundamental theorem of homomorphism to prove that

R^4/R^2 is isomorphic to R^2

Applications of linear algebra in computer science

Solve by finding the basis over R for the solution space.

(A) X + 3y -3z=0

2x - 3y + z=0

3x -2y + 2z=0

(B) X + Y + Z + W=0

2X + 3Y - Z +W=0

3X + 4Y +2W=0

A certain polynomial has a graph which has an end behaviour of II → IV. It has 2 turning points and 1 x-intercept. Sketch the shape of the graph and indicate what degree the polynomial is and whether its leading coefficient is positive or negative.