**Unformatted text preview: **3. Two vectors are said to be collinear when one can be written as a scalar multiple of the other. Consider two vectors u and v that are not collinear. Consider a vector w that does not belong to the linear span of u and v . Prove that u,v,w are linearly independent. 4. A matrix is said to be upper triangular if a ij = 0 for i > j . Consider a generic 3-by-3 upper triangular matrix A = a 11 a 12 a 13 a 22 a 23 a 33 . (a) If a 11 ,a 22 , and a 33 are nonzero, show that the only solution to Ax = 0 is x = 0. (b) If either a 11 = 0, or a 22 = 0 or a 33 = 0, then prove that the columns are linearly dependent. (Consider all three cases separately.) (c) If a 22 = 0, ﬁnd a nonzero element in the nullspace of A . 1...

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