Some problems can't be solved without the knowledge about matrices and systems of linear equations.
If a problem is solved. It is not 'the' answer.
No attempt is made to search for the most elegant answer.
I highly recommend that you at least try to solve the
problem before you read the solution.
Show that the set V = {(x,y,z)  x,y,z in R and x+y = 11} is not a subspace of R^{3} . 
Show that the set V = {(x,y,z)  x,y,z in R and x.x = z.z } is not a subspace of R^{3}. 
Show that the set V = {(x,y,z)  x,y,z in R and x + 2y + z = 0} is a subspace of R^{3}. 
Examine whether or not M = {(r,r+2,0)  r in R} is a subspace of R^{3}. 
Examine whether or not M = the set of all polynomials p(x), with p(3) = 0, is a vectorspace. 
S = {(2,5,3)} and T = {(2,0,5)} The intersection of span(S) and span(T) is a vector space. Find this space. 
Show that {(1,2,3) , (2,3,4) , (3,4,5) } is not a basis of R^{3}. 
Find a basis of R^{3} containing the vectors (1,2,5) and (0,1,2). 
Assume that v and w are linear independent vectors. Prove that v , w and (v + w) are linear dependent vectors. 
Find the coordinates of the vector (3,2,1) relative to the basis ((1,0,2),(2,1,0),(0,3,5)) in R^{3}. 
Find the solution space of the linear system
3x + 2y + 6z = 0 x  y + 2z = 0 2x + y + 8z = 0 
All polynomials p(x) with degree not greater than 2 constitute a
vector space V.
Replace in (1, 1 + x^{2} , b(x) ) the polynomial b(x) such that it becomes an ordered basis for that vector space. Calculate the coordinates of (2x^{2}  7x) relative to the chosen basis. 
S = {(2,5,3),(1,0,2)} and T = {(2,0,5),(3,5,5)} The intersection of span(S) and span(T) is a vector space. What is the dimension of that space. 
Assume that v and w are linear independent vectors.
Prove that v and (v + w) are linear independent vectors. 
Find the condition for r and s such that the vectors (r,2,s) , (r+1,2,1) and (3,s,1) are linear dependent. 
Find ,for each m, the dimension of the row space of the matrix
[2 m m1] [3 m 5 ] [1 0 m+1] 
Find ,for each m, the solution space of the linear system
3x + 2y + mz = 0 mx  y + 4z = 0 2x + y + 3z = 0 
In the vector space V = R^{3}, we take a set S = {(4,5,6) , (r,5,1) , (4,3,2)} Find the values of r such that the vector space spanned by S is not V. 
In R^{3} we have basis B = ((1,0,1) , (0,2,0) , (1,2,3)) and a basis C = ((1,0,0) , (2,0,1) , (0,0,3)) The coordinates of a vector v relative to B are (x,y,z). The coordinates of a vector v relative to C are (x',y',z'). Write the relation between these coordinates in matrix notation. 
M = span { (1+m, 4, 2); (5,6,1m) } N = span { (5+2m, 10,0) }Show that (1+m, 4, 2) and (5,6,1m) are linear independent for all real m values. Calculate m such that M+N is not a direct sum. 
Given: The vectorspace V = R^{4} is the direct sum of the spaces M and N. M = vct( (1,2,1,1) , (1,2,3,1) ) N = vct( (1,0,2,4) , (0,1,2,3) ) v = (9, 7, 9, 0)
Find:
