Linear Algebra Answers

Either prove that this statement is always true, or give a counterexample to show that it may be false: If {v1,v2,...,vp} is a linearly dependent set of vectors in R^n, and x is any vector in R^n, then {v1,v2,...,vp,x} must also be linearly dependent.

3x-3y+z=1
-x+3y+2z=-4
x+3y+5z=-7
for the above system of linear equations find a solution and show that it has infinitely many
solution or show that it has no soultion

describe the column space (range ) and the nullspace (kernel) of the matrices
A= 1 -1 B= 0 0 0
0 0 0 0 0

Linear transformation
Is this linear trandformation?
L([x,y,z]) = [0,0,0]

Thank you!

Find a 3  3-matrix A such that there exists
b1 =vector(0) and
b2 =vector(0), such that
vector(A)x = b1 has innitely many solutions, but vector(A)x = b2 has no solutions.

when x is subtracted from 2y, the difference is equal to the average of x and y What is the value of x/y?

give an example of a non-diagonalisable 4 x 4 matrix with eigenvalues -1,-1,1,1

You are given two finite dimensional subspaces of some inner product space. If one subspace is of lower dimension than the other, show whether there must exsist atleast one nonzero vector in the larger space that is orthogonal to all vectors in the smaller space.

algebraically, find the intersection points of the two lines given in each part of this problem. y=0.12x+18 y=0.08x+25

Prove that any matrix ring as vector space is direct sum of sets of symmetric and antisymmetric matrices.