1. Let a 2, 3 = -1. Let u = V = following, or explain why...

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1. Let a 2, 3 = -1. Let u = V = following, or explain why it does not make sense. (a) au + v (b) av+Bw (c) vw and w = -5 . Compute each of the 2 (d) v.w u.v 2. Let a be a scalar. Let u = (u, u2) and v = (v1, v2) be two 2-vectors. Show that (au) v = = a(u v). 3. Let u= (1,4, -6), v = (0, -3, 2) and w = (0,0,0). (a) Is w a linear combination of u and v? Justify your answer. (b) Find three nonzero vectors that are linear combinations of u and v. Justify your answer. 4. Let a, b, u, v, and z be five vectors of the same size. Suppose that: (a) u is a linear combination of a and b with coefficients P1 and p2, (b) v is a linear combination of a and b with coefficients q and 92, and z is a linear combination of u and v with coefficients a and 2. Show that z is a linear combination of a and b. (Follow the following steps) Step 1: Write the meaning of the statement in (a). Step 2: Write the meaning of the statement in (b). Step 3: Write the meaning of the statement in (c). Step 4: Express z in terms of a and b. 1 4 5. Let a 2, = -1. Let v = -1 and w = -5 1. Compute each of the following. (a) ||w|| (b) ||avw|| (c) dist(v, 2w) 1. Let a 2, 3 = -1. Let u = V = following, or explain why it does not make sense. (a) au + v (b) av+Bw (c) vw and w = -5 . Compute each of the 2 (d) v.w u.v 2. Let a be a scalar. Let u = (u, u2) and v = (v1, v2) be two 2-vectors. Show that (au) v = = a(u v). 3. Let u= (1,4, -6), v = (0, -3, 2) and w = (0,0,0). (a) Is w a linear combination of u and v? Justify your answer. (b) Find three nonzero vectors that are linear combinations of u and v. Justify your answer. 4. Let a, b, u, v, and z be five vectors of the same size. Suppose that: (a) u is a linear combination of a and b with coefficients P1 and p2, (b) v is a linear combination of a and b with coefficients q and 92, and z is a linear combination of u and v with coefficients a and 2. Show that z is a linear combination of a and b. (Follow the following steps) Step 1: Write the meaning of the statement in (a). Step 2: Write the meaning of the statement in (b). Step 3: Write the meaning of the statement in (c). Step 4: Express z in terms of a and b. 1 4 5. Let a 2, = -1. Let v = -1 and w = -5 1. Compute each of the following. (a) ||w|| (b) ||avw|| (c) dist(v, 2w)