a) A subset S of a vector space V is called a basis if
1. S is linearly independent, and
2. S is a spanning set
Any three linearly independent vectors form a basis of R3.
Let us check that whether S is a linearly independent set.
Consider the linear combination
x1⎝⎛100⎠⎞+x2⎝⎛010⎠⎞+x3⎝⎛000⎠⎞=⎝⎛000⎠⎞ This is equivalent to the matrix equation
⎝⎛100010000⎠⎞⎝⎛x1x2x3⎠⎞=⎝⎛000⎠⎞A=⎝⎛100010000000⎠⎞Thus, the general solution is x1=0,x2=0,x3=t,t∈R is a free variable.
Hence, in particular, there is a nonzero solution.
So S is linearly dependent, and hence S cannot be a basis for R3.
b)
D=⎣⎡0−203−1−191−30−11⎦⎤ Swap R1 and R2
⎣⎡−200−13−119−3−101⎦⎤
R3=R3+R2/3
⎣⎡−200−130190−101⎦⎤The rank of a matrix is the number of nonzero rows in the reduced matrix, so the rank is 3.
rank(D)=3
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