Linear Algebra Answers

If A is a hermitian matrix, then -A is skew-heritian.

True or false with full explanation

Show that T:R^3 →R^2: T(x,y,z)= (2x+y-z,x+z) is a linear transformation. Verify that T satisfies the Rank-nullity theorem

Define: R^3→R^3 by

T(x,y,z)=(x-y+z,x+y,y+z)

Let v1= (1,1,1), v2= (0,1,1), v3= (0,0,1). Find a matrix of T with respect to the basis {v1,v2,v3}. Futher check T is invertible or not.

Complete { (2, 0,3) } to form an orthogonal basis of R^3.

Write MATLAB built in functions of the following functions. Sine Cosine Tangent inverse sine inverse cosine inverse tangent Exponential diagonal components of a matrix the n by m matrix consisting of all zero the n by m matrix consisting of all one eigen values /eigen vectors of a matrix used to solve a set of linear algebraic equation

Show that if S and T are linear

transformations on a finite dimensional

vector space, then rank (ST)<= rank (S).

1.show that there are infinitely many vectors in R³ with euclidean norm 1 whose euclidean inner product with <-1,3,-5> is zero.

2.determine all values of K so that U=<-3,2k,-k> is orthogonal to V=<2,5/2,-k>.

3.find a and b such that -3ai-(-1-i) b=3a-2bi.

Express the following polynomial as a linear combination of the polynomials.

P=t²+ 4t-3

P₁=t²+2t+5

P₂=2t^2-3t

P₃=t+3

(4.3) Let ~u =< 0, 1, 1 >, ~v =< 2, 2, 0 > and w~ =< −1, 1, 0 > be three vectors in standard form.

(a) Determine which two vectors form a right angle triangle?

(b) Find θ := ~ucw~ , the angel between the given two vectors. (2)

(4.4) Let x < 0. Find the vector ~n =< x, y, z > that is orthogonal to all three vectors (2) ~u =< 1, 1, −2 >, ~v =< −1, 2, 0 > and w~ =< −1, 0, 1 >.

(4.5) Find a unit vector that is orthogonal to both ~u =< 0, −1, −1 > and ~v =< 1, 0, −1 >.

Group or not group? The set of M_nxn (R) of all nxn matrices under multiplication.