MATH 2270-1
PRACTICE EXAM 1-SP12
−x1 +2x2 +
4x4 +5x5 = −3
3x1 −7x2 +2x3 +
x5 = 4
1. Given
.
2x1 −5x2 +2x3 +4x4 +6x5 = 1
4x1 −9x2 +2x3 −4x4 −4x5 = 7
a. Write the system in vector and matrix form.
b. Given that the coefficient matrix augmented by the right hand side of the sys

1 0 −4 −28 −37 13
 0 1 −2 −12 −16 5 
tem is row equivalent to 
 then find the complete
0 0
0
0
0 0
0 0
0
0
0 0
solution of the system, and the complete solution of the related homogenous
system.


1
5 3
2. Given A =  −2
6 2 .
3 −1 1
a. Find the rank of A, and a basis for the row space of A, the column space of A,
and the null space of A.
b. Describe geometrically what the solution space to AX = 0. What then does the
solution space to AX = Y, Y 6= 0, geometrically look like assuming solutions
exist?
c. Show that N(A) is perpendicular to ROW(A).
3. Determine if the subset of vectors of R2 which lie between the two lines y = x
and y = 4x is a subspace of R2 .
4. Find a maximal number of the set of two by two matrices in
S=
A1 =
2 1
−1−1
, A2 =
1
0
−1
3
, A3 =
0
−4
12
−28
that are linearly
independent, and all linear conditions on a, b, c, d such that
a b
A=
belongs to the Span(A1 , A2 , A3 ).
c d
a. Find a basis of M2×2 that includes the independent vectors in S.
b. If A ∈ M2×2 is written as a linear combination of the matrices you found in part
(b.) with coefficients (-1,1,-1,1) then find A.