Please help me out, I need these really urgently or the course fail me.

Call a subset S of a vector space V a spanning set if Span(S) = V . Suppose that
T : V → W is a linear map of vector spaces.
a) Prove that a linear map T is 1-1 if and only if T sends linearly independent sets
to linearly independent sets.
b) Prove that T is onto if and only if T sends spanning sets to spanning sets.

 Solve the given system – or show that no solution exists:
x + 2y = 1
3x + 2y + 4z = 7
−2x + y − 2z = − 1

 Say you have k linear algebraic equations in n variables; in matrix form we write
AX = Y . Give a proof or counterexample for each of the following.
a) If n = k there is always at most one solution.
b) If n > k you can always solve AX = Y .
c) If n > k the nullspace of A has dimension greater than zero.
d) If n < k then for some Y there is no solution of AX = Y .
e) If n < k the only solution of AX = 0 is X = 0.

 Consider the system of equations
x + y − z = a
x − y + 2z = b.
a) Find the general solution of the homogeneous equation.
b) A particular solution of the inhomogeneous equations when a = 1 and b = 2
is x = 1, y = 1, z = 1. Find the most general solution of the inhomogeneous
equations.
c) Find some particular solution of the inhomogeneous equations when a = −1 and
b = −2.
d) Find some particular solution of the inhomogeneous equations when a = 3 and
b = 6.
[Remark: After you have done part a), it is possible immediately to write the solutions
to the remaining parts.]

 Consider the system of equations
x + y − z = a
x − y + 2z = b
3x + y = c
a) Find the general solution of the homogeneous equation.
b) If a = 1, b = 2, and c = 4, then a particular solution of the inhomogeneous equations is x = 1, y = 1, z = 1. Find the most general solution of these inhomogeneous
equations.
c) If a = 1, b = 2, and c = 3, show these equations have no solution.

The following questions pertain to matrices.

a) Find a 2 × 2 matrix that rotates the plane by +45 degrees (+45 degrees means 45
degrees counterclockwise).
b) Find a 2 × 2 matrix that rotates the plane by +45 degrees followed by a reflection
across the horizontal axis.
c) Find a 2 × 2 matrix that reflects across the horizontal axis followed by a rotation
the plane by +45 degrees.
d) Find a matrix that rotates the plane through +60 degrees, keeping the origin fixed.
e) Find the inverse of each of these maps.

Give a proof or counterexample the following. In each case your answers should be
brief.
a) Suppose that u, v and w are vectors in a vector space V and T : V → W is a
linear map. If u, v and w are linearly dependent, is it true that T(u), T(v) and
T(w) are linearly dependent? Why?
b) If T : R^6 → R^4 is a linear map, is it possible that the nullspace of T is one
dimensional?