I need your HELP! These are problems from my Linear Algebra class: a. Suppose th
ID: 1720844 • Letter: I
Question
I need your HELP! These are problems from my Linear Algebra class:
a. Suppose that S is a linearly independent set of vectors, and T is a subset of S, T S. Prove that T is linearly independent.
b.Suppose that T is a linearly dependent set of vectors, and T is a subset of S, T S. Prove that S is linearly dependent.
c. Suppose that {v_1, v_2, v_3, . . . , v_n} is a set of vectors. Prove that {v_1 v_2, v_2 v_3, v_3 v_4, ..., v_n v_1} is a linearly dependent set.
d. Suppose that A is an m × n matrix with linearly independent columns and the linear system LS(A, b) is consistent. Show that this system has a unique solution. (Notice that we are not requiring A to be square.)
e. Suppose that v_1 and v_2 are any two vectors from C^m. Prove the following set equality:
{v_1, v_2} = {v_1 + v_2, v_1 v_2}
Explanation / Answer
Since T is linearly independent all vectors in T cannot be represented as a linear combination of other vectors.This property holds good for S also which is a subset.
Hence S is also linearly independent.
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If T is linearly dependent there exists a linear combination for one vector in terms of the other. Hence S being a subset this property holds good and S is also linearly dependent.
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c) If v1, v2,...vn are linearly independent then n is the least integer for which this is true.
Hence even if we remove one vector they become linearly dependent.
d) Since linearly independent columns, we have independent solutions. since given consistent, they are unique.