MAT125
Exercise 5
Basis
1. a) Let,
A1 =
1
1
1
,
1
A2 =
0
1
1
,
1
A3 =
0
1
0
,
1
A4 =
0
0
0
,
1
Is the set S = A1 , A2 , A3 , A4 basis for M22 space? If not explain the reason.
b) Is the following set of vectors S basis for vector space V? Where
i)S = (1, 2), (0, 3), (1, 5)andV = R2
ii)S = (1, 3, 2), (6, 1, 1)andV = R3
If not explain the reason.
2. Find the basis and dimension of the solution space of the homogeneous linear system.
x1 − 4x2 + 3x3 − x4 = 0
2x1 − 8x2 + 6x3 − 2x4 = 0
Eigen Values & Eigen Vectors
3. Find the eigenvalues and bases for the eigenspace of A and

−2 2
A = −2 3
−4 2
A−1 where,

3
2 ,
5
Linear Combination & Independance
4. Show that, the three vectors v1 = (0, 3, 1, 1), v2 = (6, 0, 5, 1), and v3 = (4, 7, 1, 3) form a linearly dependent
set in R4 .
5. Express the following as linear combinations of u = (2, 1, 4), v = (1, 1, 3), and w = (3, 2, 5).
(a)(9, 7, 15)
(b)(6, 11, 6)
(c)(0, 0, 0)
6. Find the vector form of the general solution of the linear system Ax = b, and then use that result to find the
vector form of the general solution of Ax = 0.
x1 2x2 + x3 + 2x4 = 1
2x1 4x2 + 2x3 + 4x4 = 2
x1 + 2x2 x3 2x4 = 1
3x1 6x2 + 3x3 + 6x4 = 3