Show that the vector a=(-1,13, – 5)can be written as a linear combination of b = (1, 4, – 2) and c= (3, – 5,1).

Consider the vectors a₁ = (-1, 2, 2), a2 = (2,2, -1), and a3 = (2, -1,2). 1. Compute the projection matrices P₁ and P2 onto the lines in the direction of a₁ and a2, respectively. Multiply those projection matrices and explain why their product is what it is. 2. Find the projection vectors P1, P2, and p3 of 6 = (1,0,0) onto the lines in the direction of a1, a2, and ã3. Add the three projections p₁ + √2 + 3. What do you notice? Why does this make sense? 3. Find the projection matrix P3 onto the line directed by ã3, then find P₁ + P2 + P3. Comment on the result (explain why it makes sense).

Determine if the vectors vi = (1, –1,3, –1), v2 = (1, -1,4, 2), v3 = (1, -1, 5, 7) are linearly dependent or linearly independent.

6. a. From the set of vectors{ (1, 2, –1), (-4, –5, –6), (4, 3, 10). (5, – 3,* select two pairs of vectors that are perpendicular to each other. b. Are any of these vectors collinear? Explain.