QUESTION 1a: (18 points) Suppose R* and x, y ∈ R^3 are vectors given by
R* = [1 1 0]
x = [1 2]
y = [1 -1]
For each of the following expressions, state whether or not the expression is well-defined in the setting of linear algebra, and briefly explain your reasoning. If an expression is well defined, also calculate its value.
1. u · v
2. u × x
3. u · v + x · y
4. (u + v) × x
5. (u + v) · v
6. u × x
Recall that for any two vectors a, b ∈ R^n (of the same size), given by a = [a1 a2 ... an] and b = [b1 b2 ... bn], their dot product (a scalar) is defined as a · b = a1b1 + a2b2 + ... + anbn.
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1.
2.
3.
4.
5.
6.
QUESTION 1b: (3 points) Consider the following augmented matrix [A|b] in Reduced Row Echelon Form (RREF), representing a system of linear equations, Ax = b. Find ALL possible solutions, x, of the system, and justify your answer.
[A|b] =
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QUESTION 1c: (3 points) Consider the following augmented matrix [B|e] in Reduced Row Echelon Form (RREF), representing a system of linear equations, Bx = c. Find ALL possible solutions, x, of the system, and justify your answer.
[B|e] =
0
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