Chapter 3

Linear Dependence and Independence

[2/100%]
Show
LaTeX
Problem 3.1

For what values of the scalar kk are the three row vectors (k,1,0)(k, 1, 0), (1,k,1)(1, k, 1), and (0,1,k)(0, 1, k) linearly dependent, and for what values are they linearly independent? Describe your reasoning.

?
Problem 3.2

Let AA, BB, and CC represent three linearly independent m×nm \times n matrices. Determine whether or not the three pairwise sums A+BA+B, A+CA+C, and B+CB+C are linearly independent.

?
Remark.
?

Take advantage of the following general result on the linear dependence or independence of linear combinations: Letting A1,A2,…,AkA_1, A_2, \ldots , A_k represent m×nm \times n matrices and for j=1,…,rj = 1, \ldots , r, taking Cj=x1jA1+x2jA2+⋯+xkjAkC_j = x_{1j}A_1 + x_{2j}A_2 + \cdots + x_{kj}A_k (where x1j,x2j,…,xkjx_{1j}, x_{2j}, \ldots , x_{kj} are scalars) and letting x→j=(x1j,x2j,…,xkj)′\overrightarrow {x}_j = (x_{1j}, x_{2j}, \ldots , x_{kj})', the linear combinations C1,C2,…,CrC_1, C_2, \ldots , C_r are linearly independent if A1,A2,…,AkA_1, A_2, \ldots , A_k are linearly independent and x→1,x→2,…,x→r\overrightarrow {x}_1, \overrightarrow {x}_2, \ldots , \overrightarrow {x}_r are linearly independent, and they are linearly dependent if x→1,x→2,…,x→r\overrightarrow {x}_1, \overrightarrow {x}_2, \ldots , \overrightarrow {x}_r are linearly dependent.