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(1) The vectors A_{1} = [1, 2]^{t} and A_{2} = [2, 3]^{t} are (obviously) independent in ℝ^{2}. According to Theorem 2, they must therefore span ℝ^{2}. As an example of this, choose some random nonzero vector B in ℝ^{2} and find constants x and y such that B = xA_{1} + yA_{2}. Please do not choose either A_{1} or A_{2} as B. This is not "random"!
(3) Let A_{1} = [1, 2]^{t} and A_{2} = [2, 4]^{t} in ℝ^{2}. Find a vector B in ℝ^{2} for which there are no constants x and y, such that B = xA_{1} + yA_{2}. [Hint: Draw a picture.] Find another vector C such that x and y do exist. What geometric condition must C satisfy?
(4) "The vectors A_{1} = [1, 2]^{t} and A_{2} = [2, 3]^{t}, and A_{3} = [2, 4]^{t} span ℝ^{2}." True or false? Explain.
(5) Does the subspace W in Example 6 [[spanned by X_{1} = [1, 2, 3]^{t}, X_{2} = [3, 7, 2]^{t}]] contain Z = [9, 8, 5]^{t}?
(6) Prove that the given sets are subspaces of ℝ^{n} for the appropriate n. Find spanning sets for these spaces and find at least two different bases for each space. Give the dimension of each space. (a) S = {[a + b + 2c, 2a + b + 3c, a + b + 2c, a + 2b + 3c]^{t}  a, b, c ∈ ℝ} (b) S = {[a + 2c, 2a + b + 3c, a + b + c]^{t}  a, b, c ∈ ℝ} (c) S = {[a + b + 2c, 2a + b + 3c, a + b + c]^{t}  a, b, c ∈ ℝ}
(10) What is the dimension of M(m, n)? Prove your answer.
(15) Prove that these vectors form a basis for ℝ^{3}: X_{1} = [2, 3, 1]^{t}, X_{2} = [1, 2, 3]^{t}, X_{3} = [7, 2, 4]^{t}.
(26) Prove that the uniqueness theorem from Exercise 25 is false for spanning sets that are not bases. That is, suppose that {X_{1}, X_{2}, ..., X_{n}} spans a vector space V and is dependent. Prove that every element X ∈ V has at least two different expressions as a linear combination of the spanning vectors. It might help to consider first the case of X = 0.











