Math  /  Algebra

Question10. (a) Show that the vectors (1,0,0),(0,1,0),(0,0,1)(1,0,0),(0,1,0),(0,0,1) span R3\mathbf{R}^{3} and that they are also linearly independent. (b) Show that the vectors (1,0,0),(0,1,0),(0,0,1)(1,0,0),(0,1,0),(0,0,1), (0,1,1)(0,1,1) span R3\mathbf{R}^{3}. Demonstrate that it is not an efficient spanning set by showing that an arbitrary vector in R3\mathbf{R}^{3} can be expressed in more than one way as a linear combination of these vectors. We can think of (0,1,1)(0,1,1) as being a redundant vector. (c) Show that {(1,0,0),(0,1,0),(0,0,1),(0,1,1)}\{(1,0,0),(0,1,0),(0,0,1),(0,1,1)\} is linearly dependent and is thus not a basis for R2\mathbf{R}^{2}. A basis consists of a set of vectors, all of which are needed.

Studdy Solution
To show linear dependence, consider the set {(1,0,0),(0,1,0),(0,0,1),(0,1,1)}\{(1,0,0),(0,1,0),(0,0,1),(0,1,1)\}.
Check if there exist scalars, not all zero, such that:
a(1,0,0)+b(0,1,0)+c(0,0,1)+d(0,1,1)=(0,0,0) a(1,0,0) + b(0,1,0) + c(0,0,1) + d(0,1,1) = (0,0,0)
Choose b=db = -d and d=1d = 1, then:
a(1,0,0)+(1)(0,1,0)+c(0,0,1)+1(0,1,1)=(0,0,0) a(1,0,0) + (-1)(0,1,0) + c(0,0,1) + 1(0,1,1) = (0,0,0)
This simplifies to:
a=0,c=0,b=1,d=1 a = 0, \quad c = 0, \quad b = -1, \quad d = 1
Thus, the set is linearly dependent.

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